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|---|---|---|---|---|---|---|
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocRing α
f g : CentroidHom α
src✝ : α →+ α := ↑f - ↑g
a b : α
⊢ (⇑f - ⇑g) (a * b) = (⇑f - ⇑g) a * b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import algebra.hom.centroid from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3ad319514ff"
/-!
# Centroid homomorphisms
Let `A` be a (non unital, non associative) algebra. The centroid of `A` is the set of linear maps
`T` on `A` such that `T` commutes with left and right multiplication, that is to say, for all `a`
and `b` in `A`,
$$
T(ab) = (Ta)b, T(ab) = a(Tb).
$$
In mathlib we call elements of the centroid "centroid homomorphisms" (`CentroidHom`) in keeping
with `AddMonoidHom` etc.
We use the `FunLike` design, so each type of morphisms has a companion typeclass which is meant to
be satisfied by itself and all stricter types.
## Types of morphisms
* `CentroidHom`: Maps which preserve left and right multiplication.
## Typeclasses
* `CentroidHomClass`
## References
* [Jacobson, Structure of Rings][Jacobson1956]
* [McCrimmon, A taste of Jordan algebras][mccrimmon2004]
## Tags
centroid
-/
open Function
variable {F M N R α : Type*}
/-- The type of centroid homomorphisms from `α` to `α`. -/
structure CentroidHom (α : Type*) [NonUnitalNonAssocSemiring α] extends α →+ α where
/-- Commutativity of centroid homomorphims with left multiplication. -/
map_mul_left' (a b : α) : toFun (a * b) = a * toFun b
/-- Commutativity of centroid homomorphims with right multiplication. -/
map_mul_right' (a b : α) : toFun (a * b) = toFun a * b
#align centroid_hom CentroidHom
attribute [nolint docBlame] CentroidHom.toAddMonoidHom
/-- `CentroidHomClass F α` states that `F` is a type of centroid homomorphisms.
You should extend this class when you extend `CentroidHom`. -/
class CentroidHomClass (F : Type*) (α : outParam <| Type*) [NonUnitalNonAssocSemiring α] extends
AddMonoidHomClass F α α where
/-- Commutativity of centroid homomorphims with left multiplication. -/
map_mul_left (f : F) (a b : α) : f (a * b) = a * f b
/-- Commutativity of centroid homomorphims with right multiplication. -/
map_mul_right (f : F) (a b : α) : f (a * b) = f a * b
#align centroid_hom_class CentroidHomClass
export CentroidHomClass (map_mul_left map_mul_right)
instance [NonUnitalNonAssocSemiring α] [CentroidHomClass F α] : CoeTC F (CentroidHom α) :=
⟨fun f ↦
{ (f : α →+ α) with
toFun := f
map_mul_left' := map_mul_left f
map_mul_right' := map_mul_right f }⟩
/-! ### Centroid homomorphisms -/
namespace CentroidHom
section NonUnitalNonAssocSemiring
variable [NonUnitalNonAssocSemiring α]
instance : CentroidHomClass (CentroidHom α) α where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
congr with x
exact congrFun h x
map_zero f := f.map_zero'
map_add f := f.map_add'
map_mul_left f := f.map_mul_left'
map_mul_right f := f.map_mul_right'
/-- Helper instance for when there's too many metavariables to apply `FunLike.CoeFun`
directly. -/
/- Porting note: Lean gave me `unknown constant 'FunLike.CoeFun'` and says `CoeFun` is a type
mismatch, so I used `library_search`. -/
instance : CoeFun (CentroidHom α) fun _ ↦ α → α :=
inferInstanceAs (CoeFun (CentroidHom α) fun _ ↦ α → α)
-- Porting note: removed @[simp]; not in normal form. (`toAddMonoidHom_eq_coe` below ensures that
-- the LHS simplifies to the RHS anyway.)
theorem toFun_eq_coe {f : CentroidHom α} : f.toFun = f := rfl
#align centroid_hom.to_fun_eq_coe CentroidHom.toFun_eq_coe
@[ext]
theorem ext {f g : CentroidHom α} (h : ∀ a, f a = g a) : f = g :=
FunLike.ext f g h
#align centroid_hom.ext CentroidHom.ext
@[simp, norm_cast]
theorem coe_toAddMonoidHom (f : CentroidHom α) : ⇑(f : α →+ α) = f :=
rfl
#align centroid_hom.coe_to_add_monoid_hom CentroidHom.coe_toAddMonoidHom
@[simp]
theorem toAddMonoidHom_eq_coe (f : CentroidHom α) : f.toAddMonoidHom = f :=
rfl
#align centroid_hom.to_add_monoid_hom_eq_coe CentroidHom.toAddMonoidHom_eq_coe
theorem coe_toAddMonoidHom_injective : Injective ((↑) : CentroidHom α → α →+ α) :=
fun _f _g h => ext fun a ↦
haveI := FunLike.congr_fun h a
this
#align centroid_hom.coe_to_add_monoid_hom_injective CentroidHom.coe_toAddMonoidHom_injective
/-- Turn a centroid homomorphism into an additive monoid endomorphism. -/
def toEnd (f : CentroidHom α) : AddMonoid.End α :=
(f : α →+ α)
#align centroid_hom.to_End CentroidHom.toEnd
theorem toEnd_injective : Injective (CentroidHom.toEnd : CentroidHom α → AddMonoid.End α) :=
coe_toAddMonoidHom_injective
#align centroid_hom.to_End_injective CentroidHom.toEnd_injective
/-- Copy of a `CentroidHom` with a new `toFun` equal to the old one. Useful to fix
definitional equalities. -/
protected def copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : CentroidHom α :=
{ f.toAddMonoidHom.copy f' <| h with
toFun := f'
map_mul_left' := fun a b ↦ by simp_rw [h, map_mul_left]
map_mul_right' := fun a b ↦ by simp_rw [h, map_mul_right] }
#align centroid_hom.copy CentroidHom.copy
@[simp]
theorem coe_copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : ⇑(f.copy f' h) = f' :=
rfl
#align centroid_hom.coe_copy CentroidHom.coe_copy
theorem copy_eq (f : CentroidHom α) (f' : α → α) (h : f' = f) : f.copy f' h = f :=
FunLike.ext' h
#align centroid_hom.copy_eq CentroidHom.copy_eq
variable (α)
/-- `id` as a `CentroidHom`. -/
protected def id : CentroidHom α :=
{ AddMonoidHom.id α with
map_mul_left' := fun _ _ ↦ rfl
map_mul_right' := fun _ _ ↦ rfl }
#align centroid_hom.id CentroidHom.id
instance : Inhabited (CentroidHom α) :=
⟨CentroidHom.id α⟩
@[simp, norm_cast]
theorem coe_id : ⇑(CentroidHom.id α) = id :=
rfl
#align centroid_hom.coe_id CentroidHom.coe_id
@[simp, norm_cast]
theorem toAddMonoidHom_id : (CentroidHom.id α : α →+ α) = AddMonoidHom.id α :=
rfl
#align centroid_hom.coe_to_add_monoid_hom_id CentroidHom.toAddMonoidHom_id
variable {α}
@[simp]
theorem id_apply (a : α) : CentroidHom.id α a = a :=
rfl
#align centroid_hom.id_apply CentroidHom.id_apply
/-- Composition of `CentroidHom`s as a `CentroidHom`. -/
def comp (g f : CentroidHom α) : CentroidHom α :=
{ g.toAddMonoidHom.comp f.toAddMonoidHom with
map_mul_left' := fun _a _b ↦ (congr_arg g <| f.map_mul_left' _ _).trans <| g.map_mul_left' _ _
map_mul_right' := fun _a _b ↦
(congr_arg g <| f.map_mul_right' _ _).trans <| g.map_mul_right' _ _ }
#align centroid_hom.comp CentroidHom.comp
@[simp, norm_cast]
theorem coe_comp (g f : CentroidHom α) : ⇑(g.comp f) = g ∘ f :=
rfl
#align centroid_hom.coe_comp CentroidHom.coe_comp
@[simp]
theorem comp_apply (g f : CentroidHom α) (a : α) : g.comp f a = g (f a) :=
rfl
#align centroid_hom.comp_apply CentroidHom.comp_apply
@[simp, norm_cast]
theorem coe_comp_addMonoidHom (g f : CentroidHom α) : (g.comp f : α →+ α) = (g : α →+ α).comp f :=
rfl
#align centroid_hom.coe_comp_add_monoid_hom CentroidHom.coe_comp_addMonoidHom
@[simp]
theorem comp_assoc (h g f : CentroidHom α) : (h.comp g).comp f = h.comp (g.comp f) :=
rfl
#align centroid_hom.comp_assoc CentroidHom.comp_assoc
@[simp]
theorem comp_id (f : CentroidHom α) : f.comp (CentroidHom.id α) = f :=
rfl
#align centroid_hom.comp_id CentroidHom.comp_id
@[simp]
theorem id_comp (f : CentroidHom α) : (CentroidHom.id α).comp f = f :=
rfl
#align centroid_hom.id_comp CentroidHom.id_comp
@[simp]
theorem cancel_right {g₁ g₂ f : CentroidHom α} (hf : Surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨fun h ↦ ext <| hf.forall.2 <| FunLike.ext_iff.1 h, fun a ↦ congrFun (congrArg comp a) f⟩
#align centroid_hom.cancel_right CentroidHom.cancel_right
@[simp]
theorem cancel_left {g f₁ f₂ : CentroidHom α} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨fun h ↦ ext fun a ↦ hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩
#align centroid_hom.cancel_left CentroidHom.cancel_left
instance : Zero (CentroidHom α) :=
⟨{ (0 : α →+ α) with
map_mul_left' := fun _a _b ↦ (mul_zero _).symm
map_mul_right' := fun _a _b ↦ (zero_mul _).symm }⟩
instance : One (CentroidHom α) :=
⟨CentroidHom.id α⟩
instance : Add (CentroidHom α) :=
⟨fun f g ↦
{ (f + g : α →+ α) with
map_mul_left' := fun a b ↦ by
show f (a * b) + g (a * b) = a * (f b + g b)
simp [map_mul_left, mul_add]
map_mul_right' := fun a b ↦ by
show f (a * b) + g (a * b) = (f a + g a) * b
simp [map_mul_right, add_mul] }⟩
instance : Mul (CentroidHom α) :=
⟨comp⟩
variable [Monoid M] [Monoid N] [Semiring R]
variable [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α]
variable [DistribMulAction N α] [SMulCommClass N α α] [IsScalarTower N α α]
variable [Module R α] [SMulCommClass R α α] [IsScalarTower R α α]
instance instSMul : SMul M (CentroidHom α) where
smul n f :=
{ (n • f : α →+ α) with
map_mul_left' := fun a b ↦ by
change n • f (a * b) = a * n • f b
rw [map_mul_left f, ← mul_smul_comm]
map_mul_right' := fun a b ↦ by
change n • f (a * b) = n • f a * b
rw [map_mul_right f, ← smul_mul_assoc] }
#noalign centroid_hom.has_nsmul
instance [SMul M N] [IsScalarTower M N α] : IsScalarTower M N (CentroidHom α) where
smul_assoc _ _ _ := ext <| fun _ => smul_assoc _ _ _
instance [SMulCommClass M N α] : SMulCommClass M N (CentroidHom α) where
smul_comm _ _ _ := ext <| fun _ => smul_comm _ _ _
instance [DistribMulAction Mᵐᵒᵖ α] [IsCentralScalar M α] : IsCentralScalar M (CentroidHom α) where
op_smul_eq_smul _ _ := ext <| fun _ => op_smul_eq_smul _ _
instance isScalarTowerRight : IsScalarTower M (CentroidHom α) (CentroidHom α) where
smul_assoc _ _ _ := rfl
instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_left' _ _)
map_mul_right' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_right' _ _) }⟩
#align centroid_hom.has_npow_nat CentroidHom.hasNPowNat
@[simp, norm_cast]
theorem coe_zero : ⇑(0 : CentroidHom α) = 0 :=
rfl
#align centroid_hom.coe_zero CentroidHom.coe_zero
@[simp, norm_cast]
theorem coe_one : ⇑(1 : CentroidHom α) = id :=
rfl
#align centroid_hom.coe_one CentroidHom.coe_one
@[simp, norm_cast]
theorem coe_add (f g : CentroidHom α) : ⇑(f + g) = f + g :=
rfl
#align centroid_hom.coe_add CentroidHom.coe_add
@[simp, norm_cast]
theorem coe_mul (f g : CentroidHom α) : ⇑(f * g) = f ∘ g :=
rfl
#align centroid_hom.coe_mul CentroidHom.coe_mul
@[simp, norm_cast]
theorem coe_smul (n : M) (f : CentroidHom α) : ⇑(n • f) = n • ⇑f :=
rfl
#align centroid_hom.coe_nsmul CentroidHom.coe_smul
@[simp]
theorem zero_apply (a : α) : (0 : CentroidHom α) a = 0 :=
rfl
#align centroid_hom.zero_apply CentroidHom.zero_apply
@[simp]
theorem one_apply (a : α) : (1 : CentroidHom α) a = a :=
rfl
#align centroid_hom.one_apply CentroidHom.one_apply
@[simp]
theorem add_apply (f g : CentroidHom α) (a : α) : (f + g) a = f a + g a :=
rfl
#align centroid_hom.add_apply CentroidHom.add_apply
@[simp]
theorem mul_apply (f g : CentroidHom α) (a : α) : (f * g) a = f (g a) :=
rfl
#align centroid_hom.mul_apply CentroidHom.mul_apply
@[simp]
theorem smul_apply (n : M) (f : CentroidHom α) (a : α) : (n • f) a = n • f a :=
rfl
#align centroid_hom.nsmul_apply CentroidHom.smul_apply
example : SMul ℕ (CentroidHom α) := instSMul
@[simp]
theorem toEnd_zero : (0 : CentroidHom α).toEnd = 0 :=
rfl
#align centroid_hom.to_End_zero CentroidHom.toEnd_zero
@[simp]
theorem toEnd_add (x y : CentroidHom α) : (x + y).toEnd = x.toEnd + y.toEnd :=
rfl
#align centroid_hom.to_End_add CentroidHom.toEnd_add
theorem toEnd_smul (m : M) (x : CentroidHom α) : (m • x).toEnd = m • x.toEnd :=
rfl
#align centroid_hom.to_End_nsmul CentroidHom.toEnd_smul
instance : AddCommMonoid (CentroidHom α) :=
coe_toAddMonoidHom_injective.addCommMonoid _ toEnd_zero toEnd_add (swap toEnd_smul)
instance : NatCast (CentroidHom α) where natCast n := n • (1 : CentroidHom α)
-- Porting note: `nolint simpNF` added because simplify fails on left-hand side
@[simp, norm_cast, nolint simpNF]
theorem coe_nat_cast (n : ℕ) : ⇑(n : CentroidHom α) = n • (CentroidHom.id α) :=
rfl
#align centroid_hom.coe_nat_cast CentroidHom.coe_nat_cast
theorem nat_cast_apply (n : ℕ) (m : α) : (n : CentroidHom α) m = n • m :=
rfl
#align centroid_hom.nat_cast_apply CentroidHom.nat_cast_apply
@[simp]
theorem toEnd_one : (1 : CentroidHom α).toEnd = 1 :=
rfl
#align centroid_hom.to_End_one CentroidHom.toEnd_one
@[simp]
theorem toEnd_mul (x y : CentroidHom α) : (x * y).toEnd = x.toEnd * y.toEnd :=
rfl
#align centroid_hom.to_End_mul CentroidHom.toEnd_mul
@[simp]
theorem toEnd_pow (x : CentroidHom α) (n : ℕ) : (x ^ n).toEnd = x.toEnd ^ n :=
rfl
#align centroid_hom.to_End_pow CentroidHom.toEnd_pow
@[simp, norm_cast]
theorem toEnd_nat_cast (n : ℕ) : (n : CentroidHom α).toEnd = ↑n :=
rfl
#align centroid_hom.to_End_nat_cast CentroidHom.toEnd_nat_cast
-- cf `add_monoid.End.semiring`
instance : Semiring (CentroidHom α) :=
toEnd_injective.semiring _ toEnd_zero toEnd_one toEnd_add toEnd_mul (swap toEnd_smul) toEnd_pow
toEnd_nat_cast
variable (α) in
/-- `CentroidHom.toEnd` as a `RingHom`. -/
@[simps]
def toEndRingHom : CentroidHom α →+* AddMonoid.End α where
toFun := toEnd
map_zero' := toEnd_zero
map_one' := toEnd_one
map_add' := toEnd_add
map_mul' := toEnd_mul
theorem comp_mul_comm (T S : CentroidHom α) (a b : α) : (T ∘ S) (a * b) = (S ∘ T) (a * b) := by
simp only [Function.comp_apply]
rw [map_mul_right, map_mul_left, ← map_mul_right, ← map_mul_left]
#align centroid_hom.comp_mul_comm CentroidHom.comp_mul_comm
instance : DistribMulAction M (CentroidHom α) :=
toEnd_injective.distribMulAction (toEndRingHom α).toAddMonoidHom toEnd_smul
instance : Module R (CentroidHom α) :=
toEnd_injective.module R (toEndRingHom α).toAddMonoidHom toEnd_smul
local notation "L" => AddMonoid.End.mulLeft
local notation "R" => AddMonoid.End.mulRight
lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
refine ⟨?_, fun h ↦ ?_⟩
· rintro ⟨f, rfl⟩ S (⟨a, rfl⟩ | ⟨b, rfl⟩)
· exact AddMonoidHom.ext fun b ↦ (map_mul_left f a b).symm
· exact AddMonoidHom.ext fun a ↦ (map_mul_right f a b).symm
· rw [Subsemiring.mem_centralizer_iff] at h
refine ⟨⟨T, fun a b ↦ ?_, fun a b ↦ ?_⟩, rfl⟩
· exact congr($(h (L a) (.inl ⟨a, rfl⟩)) b).symm
· exact congr($(h (R b) (.inr ⟨b, rfl⟩)) a).symm
/-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z :=
{ L (z : α) with
map_mul_left' := ((Set.mem_center_iff _).mp z.prop).left_comm
map_mul_right' := ((Set.mem_center_iff _).mp z.prop).left_assoc }
map_zero' := by
simp only [ZeroMemClass.coe_zero, map_zero]
exact rfl
map_add' := fun _ _ => by
simp only [AddSubmonoid.coe_add, NonUnitalSubsemiring.coe_toAddSubmonoid, map_add]
exact rfl
map_mul' := fun z₁ z₂ => by
ext a
exact (((Set.mem_center_iff _).mp z₁.prop).left_assoc z₂ a).symm
lemma centerToCentroid_apply (z : { x // x ∈ NonUnitalSubsemiring.center α }) (a : α) :
(centerToCentroid z) a = z * a := rfl
lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
have e2 (d : α) : T d = d * a := congr($(hT.trans hc) d)
constructor
case comm => exact (congr($hc ·))
case left_assoc => simpa [e1] using (map_mul_right T · ·)
case mid_assoc => exact fun b c ↦ by simpa [e1 c, e2 b] using
(map_mul_right T b c).symm.trans <| map_mul_left T b c
case right_assoc => simpa [e2] using (map_mul_left T · ·)
end NonUnitalNonAssocSemiring
section NonAssocSemiring
variable [NonAssocSemiring α]
/-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by refine ⟨?_, ?_, ?_, ?_⟩; all_goals simp [← map_mul_left, ← map_mul_right]⟩
left_inv := fun z ↦ Subtype.ext <| by simp [centerToCentroid_apply]
right_inv := fun T ↦ CentroidHom.ext <| by simp [centerToCentroid_apply, ← map_mul_right] }
end NonAssocSemiring
section NonUnitalNonAssocRing
variable [NonUnitalNonAssocRing α]
/-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) :=
⟨fun f ↦
{ (-f : α →+ α) with
map_mul_left' := fun a b ↦ by
change -f (a * b) = a * (-f b)
simp [map_mul_left]
map_mul_right' := fun a b ↦ by
change -f (a * b) = (-f a) * b
simp [map_mul_right] }⟩
instance : Sub (CentroidHom α) :=
⟨fun f g ↦
{ (f - g : α →+ α) with
map_mul_left' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = a * (⇑f - ⇑g) b
simp [map_mul_left, mul_sub]
map_mul_right' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = ((⇑f - ⇑g) a) * b
| simp [map_mul_right, sub_mul] | instance : Sub (CentroidHom α) :=
⟨fun f g ↦
{ (f - g : α →+ α) with
map_mul_left' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = a * (⇑f - ⇑g) b
simp [map_mul_left, mul_sub]
map_mul_right' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = ((⇑f - ⇑g) a) * b
| Mathlib.Algebra.Ring.CentroidHom.514_0.FQQ3LT1tg3cKlkH | instance : Sub (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalRing α
h : ∀ (a b : α), (∀ (r : α), a * r * b = 0) → a = 0 ∨ b = 0
src✝ : Ring (CentroidHom α) := instRing
f g : CentroidHom α
⊢ f * g = g * f | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import algebra.hom.centroid from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3ad319514ff"
/-!
# Centroid homomorphisms
Let `A` be a (non unital, non associative) algebra. The centroid of `A` is the set of linear maps
`T` on `A` such that `T` commutes with left and right multiplication, that is to say, for all `a`
and `b` in `A`,
$$
T(ab) = (Ta)b, T(ab) = a(Tb).
$$
In mathlib we call elements of the centroid "centroid homomorphisms" (`CentroidHom`) in keeping
with `AddMonoidHom` etc.
We use the `FunLike` design, so each type of morphisms has a companion typeclass which is meant to
be satisfied by itself and all stricter types.
## Types of morphisms
* `CentroidHom`: Maps which preserve left and right multiplication.
## Typeclasses
* `CentroidHomClass`
## References
* [Jacobson, Structure of Rings][Jacobson1956]
* [McCrimmon, A taste of Jordan algebras][mccrimmon2004]
## Tags
centroid
-/
open Function
variable {F M N R α : Type*}
/-- The type of centroid homomorphisms from `α` to `α`. -/
structure CentroidHom (α : Type*) [NonUnitalNonAssocSemiring α] extends α →+ α where
/-- Commutativity of centroid homomorphims with left multiplication. -/
map_mul_left' (a b : α) : toFun (a * b) = a * toFun b
/-- Commutativity of centroid homomorphims with right multiplication. -/
map_mul_right' (a b : α) : toFun (a * b) = toFun a * b
#align centroid_hom CentroidHom
attribute [nolint docBlame] CentroidHom.toAddMonoidHom
/-- `CentroidHomClass F α` states that `F` is a type of centroid homomorphisms.
You should extend this class when you extend `CentroidHom`. -/
class CentroidHomClass (F : Type*) (α : outParam <| Type*) [NonUnitalNonAssocSemiring α] extends
AddMonoidHomClass F α α where
/-- Commutativity of centroid homomorphims with left multiplication. -/
map_mul_left (f : F) (a b : α) : f (a * b) = a * f b
/-- Commutativity of centroid homomorphims with right multiplication. -/
map_mul_right (f : F) (a b : α) : f (a * b) = f a * b
#align centroid_hom_class CentroidHomClass
export CentroidHomClass (map_mul_left map_mul_right)
instance [NonUnitalNonAssocSemiring α] [CentroidHomClass F α] : CoeTC F (CentroidHom α) :=
⟨fun f ↦
{ (f : α →+ α) with
toFun := f
map_mul_left' := map_mul_left f
map_mul_right' := map_mul_right f }⟩
/-! ### Centroid homomorphisms -/
namespace CentroidHom
section NonUnitalNonAssocSemiring
variable [NonUnitalNonAssocSemiring α]
instance : CentroidHomClass (CentroidHom α) α where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
congr with x
exact congrFun h x
map_zero f := f.map_zero'
map_add f := f.map_add'
map_mul_left f := f.map_mul_left'
map_mul_right f := f.map_mul_right'
/-- Helper instance for when there's too many metavariables to apply `FunLike.CoeFun`
directly. -/
/- Porting note: Lean gave me `unknown constant 'FunLike.CoeFun'` and says `CoeFun` is a type
mismatch, so I used `library_search`. -/
instance : CoeFun (CentroidHom α) fun _ ↦ α → α :=
inferInstanceAs (CoeFun (CentroidHom α) fun _ ↦ α → α)
-- Porting note: removed @[simp]; not in normal form. (`toAddMonoidHom_eq_coe` below ensures that
-- the LHS simplifies to the RHS anyway.)
theorem toFun_eq_coe {f : CentroidHom α} : f.toFun = f := rfl
#align centroid_hom.to_fun_eq_coe CentroidHom.toFun_eq_coe
@[ext]
theorem ext {f g : CentroidHom α} (h : ∀ a, f a = g a) : f = g :=
FunLike.ext f g h
#align centroid_hom.ext CentroidHom.ext
@[simp, norm_cast]
theorem coe_toAddMonoidHom (f : CentroidHom α) : ⇑(f : α →+ α) = f :=
rfl
#align centroid_hom.coe_to_add_monoid_hom CentroidHom.coe_toAddMonoidHom
@[simp]
theorem toAddMonoidHom_eq_coe (f : CentroidHom α) : f.toAddMonoidHom = f :=
rfl
#align centroid_hom.to_add_monoid_hom_eq_coe CentroidHom.toAddMonoidHom_eq_coe
theorem coe_toAddMonoidHom_injective : Injective ((↑) : CentroidHom α → α →+ α) :=
fun _f _g h => ext fun a ↦
haveI := FunLike.congr_fun h a
this
#align centroid_hom.coe_to_add_monoid_hom_injective CentroidHom.coe_toAddMonoidHom_injective
/-- Turn a centroid homomorphism into an additive monoid endomorphism. -/
def toEnd (f : CentroidHom α) : AddMonoid.End α :=
(f : α →+ α)
#align centroid_hom.to_End CentroidHom.toEnd
theorem toEnd_injective : Injective (CentroidHom.toEnd : CentroidHom α → AddMonoid.End α) :=
coe_toAddMonoidHom_injective
#align centroid_hom.to_End_injective CentroidHom.toEnd_injective
/-- Copy of a `CentroidHom` with a new `toFun` equal to the old one. Useful to fix
definitional equalities. -/
protected def copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : CentroidHom α :=
{ f.toAddMonoidHom.copy f' <| h with
toFun := f'
map_mul_left' := fun a b ↦ by simp_rw [h, map_mul_left]
map_mul_right' := fun a b ↦ by simp_rw [h, map_mul_right] }
#align centroid_hom.copy CentroidHom.copy
@[simp]
theorem coe_copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : ⇑(f.copy f' h) = f' :=
rfl
#align centroid_hom.coe_copy CentroidHom.coe_copy
theorem copy_eq (f : CentroidHom α) (f' : α → α) (h : f' = f) : f.copy f' h = f :=
FunLike.ext' h
#align centroid_hom.copy_eq CentroidHom.copy_eq
variable (α)
/-- `id` as a `CentroidHom`. -/
protected def id : CentroidHom α :=
{ AddMonoidHom.id α with
map_mul_left' := fun _ _ ↦ rfl
map_mul_right' := fun _ _ ↦ rfl }
#align centroid_hom.id CentroidHom.id
instance : Inhabited (CentroidHom α) :=
⟨CentroidHom.id α⟩
@[simp, norm_cast]
theorem coe_id : ⇑(CentroidHom.id α) = id :=
rfl
#align centroid_hom.coe_id CentroidHom.coe_id
@[simp, norm_cast]
theorem toAddMonoidHom_id : (CentroidHom.id α : α →+ α) = AddMonoidHom.id α :=
rfl
#align centroid_hom.coe_to_add_monoid_hom_id CentroidHom.toAddMonoidHom_id
variable {α}
@[simp]
theorem id_apply (a : α) : CentroidHom.id α a = a :=
rfl
#align centroid_hom.id_apply CentroidHom.id_apply
/-- Composition of `CentroidHom`s as a `CentroidHom`. -/
def comp (g f : CentroidHom α) : CentroidHom α :=
{ g.toAddMonoidHom.comp f.toAddMonoidHom with
map_mul_left' := fun _a _b ↦ (congr_arg g <| f.map_mul_left' _ _).trans <| g.map_mul_left' _ _
map_mul_right' := fun _a _b ↦
(congr_arg g <| f.map_mul_right' _ _).trans <| g.map_mul_right' _ _ }
#align centroid_hom.comp CentroidHom.comp
@[simp, norm_cast]
theorem coe_comp (g f : CentroidHom α) : ⇑(g.comp f) = g ∘ f :=
rfl
#align centroid_hom.coe_comp CentroidHom.coe_comp
@[simp]
theorem comp_apply (g f : CentroidHom α) (a : α) : g.comp f a = g (f a) :=
rfl
#align centroid_hom.comp_apply CentroidHom.comp_apply
@[simp, norm_cast]
theorem coe_comp_addMonoidHom (g f : CentroidHom α) : (g.comp f : α →+ α) = (g : α →+ α).comp f :=
rfl
#align centroid_hom.coe_comp_add_monoid_hom CentroidHom.coe_comp_addMonoidHom
@[simp]
theorem comp_assoc (h g f : CentroidHom α) : (h.comp g).comp f = h.comp (g.comp f) :=
rfl
#align centroid_hom.comp_assoc CentroidHom.comp_assoc
@[simp]
theorem comp_id (f : CentroidHom α) : f.comp (CentroidHom.id α) = f :=
rfl
#align centroid_hom.comp_id CentroidHom.comp_id
@[simp]
theorem id_comp (f : CentroidHom α) : (CentroidHom.id α).comp f = f :=
rfl
#align centroid_hom.id_comp CentroidHom.id_comp
@[simp]
theorem cancel_right {g₁ g₂ f : CentroidHom α} (hf : Surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨fun h ↦ ext <| hf.forall.2 <| FunLike.ext_iff.1 h, fun a ↦ congrFun (congrArg comp a) f⟩
#align centroid_hom.cancel_right CentroidHom.cancel_right
@[simp]
theorem cancel_left {g f₁ f₂ : CentroidHom α} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨fun h ↦ ext fun a ↦ hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩
#align centroid_hom.cancel_left CentroidHom.cancel_left
instance : Zero (CentroidHom α) :=
⟨{ (0 : α →+ α) with
map_mul_left' := fun _a _b ↦ (mul_zero _).symm
map_mul_right' := fun _a _b ↦ (zero_mul _).symm }⟩
instance : One (CentroidHom α) :=
⟨CentroidHom.id α⟩
instance : Add (CentroidHom α) :=
⟨fun f g ↦
{ (f + g : α →+ α) with
map_mul_left' := fun a b ↦ by
show f (a * b) + g (a * b) = a * (f b + g b)
simp [map_mul_left, mul_add]
map_mul_right' := fun a b ↦ by
show f (a * b) + g (a * b) = (f a + g a) * b
simp [map_mul_right, add_mul] }⟩
instance : Mul (CentroidHom α) :=
⟨comp⟩
variable [Monoid M] [Monoid N] [Semiring R]
variable [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α]
variable [DistribMulAction N α] [SMulCommClass N α α] [IsScalarTower N α α]
variable [Module R α] [SMulCommClass R α α] [IsScalarTower R α α]
instance instSMul : SMul M (CentroidHom α) where
smul n f :=
{ (n • f : α →+ α) with
map_mul_left' := fun a b ↦ by
change n • f (a * b) = a * n • f b
rw [map_mul_left f, ← mul_smul_comm]
map_mul_right' := fun a b ↦ by
change n • f (a * b) = n • f a * b
rw [map_mul_right f, ← smul_mul_assoc] }
#noalign centroid_hom.has_nsmul
instance [SMul M N] [IsScalarTower M N α] : IsScalarTower M N (CentroidHom α) where
smul_assoc _ _ _ := ext <| fun _ => smul_assoc _ _ _
instance [SMulCommClass M N α] : SMulCommClass M N (CentroidHom α) where
smul_comm _ _ _ := ext <| fun _ => smul_comm _ _ _
instance [DistribMulAction Mᵐᵒᵖ α] [IsCentralScalar M α] : IsCentralScalar M (CentroidHom α) where
op_smul_eq_smul _ _ := ext <| fun _ => op_smul_eq_smul _ _
instance isScalarTowerRight : IsScalarTower M (CentroidHom α) (CentroidHom α) where
smul_assoc _ _ _ := rfl
instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_left' _ _)
map_mul_right' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_right' _ _) }⟩
#align centroid_hom.has_npow_nat CentroidHom.hasNPowNat
@[simp, norm_cast]
theorem coe_zero : ⇑(0 : CentroidHom α) = 0 :=
rfl
#align centroid_hom.coe_zero CentroidHom.coe_zero
@[simp, norm_cast]
theorem coe_one : ⇑(1 : CentroidHom α) = id :=
rfl
#align centroid_hom.coe_one CentroidHom.coe_one
@[simp, norm_cast]
theorem coe_add (f g : CentroidHom α) : ⇑(f + g) = f + g :=
rfl
#align centroid_hom.coe_add CentroidHom.coe_add
@[simp, norm_cast]
theorem coe_mul (f g : CentroidHom α) : ⇑(f * g) = f ∘ g :=
rfl
#align centroid_hom.coe_mul CentroidHom.coe_mul
@[simp, norm_cast]
theorem coe_smul (n : M) (f : CentroidHom α) : ⇑(n • f) = n • ⇑f :=
rfl
#align centroid_hom.coe_nsmul CentroidHom.coe_smul
@[simp]
theorem zero_apply (a : α) : (0 : CentroidHom α) a = 0 :=
rfl
#align centroid_hom.zero_apply CentroidHom.zero_apply
@[simp]
theorem one_apply (a : α) : (1 : CentroidHom α) a = a :=
rfl
#align centroid_hom.one_apply CentroidHom.one_apply
@[simp]
theorem add_apply (f g : CentroidHom α) (a : α) : (f + g) a = f a + g a :=
rfl
#align centroid_hom.add_apply CentroidHom.add_apply
@[simp]
theorem mul_apply (f g : CentroidHom α) (a : α) : (f * g) a = f (g a) :=
rfl
#align centroid_hom.mul_apply CentroidHom.mul_apply
@[simp]
theorem smul_apply (n : M) (f : CentroidHom α) (a : α) : (n • f) a = n • f a :=
rfl
#align centroid_hom.nsmul_apply CentroidHom.smul_apply
example : SMul ℕ (CentroidHom α) := instSMul
@[simp]
theorem toEnd_zero : (0 : CentroidHom α).toEnd = 0 :=
rfl
#align centroid_hom.to_End_zero CentroidHom.toEnd_zero
@[simp]
theorem toEnd_add (x y : CentroidHom α) : (x + y).toEnd = x.toEnd + y.toEnd :=
rfl
#align centroid_hom.to_End_add CentroidHom.toEnd_add
theorem toEnd_smul (m : M) (x : CentroidHom α) : (m • x).toEnd = m • x.toEnd :=
rfl
#align centroid_hom.to_End_nsmul CentroidHom.toEnd_smul
instance : AddCommMonoid (CentroidHom α) :=
coe_toAddMonoidHom_injective.addCommMonoid _ toEnd_zero toEnd_add (swap toEnd_smul)
instance : NatCast (CentroidHom α) where natCast n := n • (1 : CentroidHom α)
-- Porting note: `nolint simpNF` added because simplify fails on left-hand side
@[simp, norm_cast, nolint simpNF]
theorem coe_nat_cast (n : ℕ) : ⇑(n : CentroidHom α) = n • (CentroidHom.id α) :=
rfl
#align centroid_hom.coe_nat_cast CentroidHom.coe_nat_cast
theorem nat_cast_apply (n : ℕ) (m : α) : (n : CentroidHom α) m = n • m :=
rfl
#align centroid_hom.nat_cast_apply CentroidHom.nat_cast_apply
@[simp]
theorem toEnd_one : (1 : CentroidHom α).toEnd = 1 :=
rfl
#align centroid_hom.to_End_one CentroidHom.toEnd_one
@[simp]
theorem toEnd_mul (x y : CentroidHom α) : (x * y).toEnd = x.toEnd * y.toEnd :=
rfl
#align centroid_hom.to_End_mul CentroidHom.toEnd_mul
@[simp]
theorem toEnd_pow (x : CentroidHom α) (n : ℕ) : (x ^ n).toEnd = x.toEnd ^ n :=
rfl
#align centroid_hom.to_End_pow CentroidHom.toEnd_pow
@[simp, norm_cast]
theorem toEnd_nat_cast (n : ℕ) : (n : CentroidHom α).toEnd = ↑n :=
rfl
#align centroid_hom.to_End_nat_cast CentroidHom.toEnd_nat_cast
-- cf `add_monoid.End.semiring`
instance : Semiring (CentroidHom α) :=
toEnd_injective.semiring _ toEnd_zero toEnd_one toEnd_add toEnd_mul (swap toEnd_smul) toEnd_pow
toEnd_nat_cast
variable (α) in
/-- `CentroidHom.toEnd` as a `RingHom`. -/
@[simps]
def toEndRingHom : CentroidHom α →+* AddMonoid.End α where
toFun := toEnd
map_zero' := toEnd_zero
map_one' := toEnd_one
map_add' := toEnd_add
map_mul' := toEnd_mul
theorem comp_mul_comm (T S : CentroidHom α) (a b : α) : (T ∘ S) (a * b) = (S ∘ T) (a * b) := by
simp only [Function.comp_apply]
rw [map_mul_right, map_mul_left, ← map_mul_right, ← map_mul_left]
#align centroid_hom.comp_mul_comm CentroidHom.comp_mul_comm
instance : DistribMulAction M (CentroidHom α) :=
toEnd_injective.distribMulAction (toEndRingHom α).toAddMonoidHom toEnd_smul
instance : Module R (CentroidHom α) :=
toEnd_injective.module R (toEndRingHom α).toAddMonoidHom toEnd_smul
local notation "L" => AddMonoid.End.mulLeft
local notation "R" => AddMonoid.End.mulRight
lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
refine ⟨?_, fun h ↦ ?_⟩
· rintro ⟨f, rfl⟩ S (⟨a, rfl⟩ | ⟨b, rfl⟩)
· exact AddMonoidHom.ext fun b ↦ (map_mul_left f a b).symm
· exact AddMonoidHom.ext fun a ↦ (map_mul_right f a b).symm
· rw [Subsemiring.mem_centralizer_iff] at h
refine ⟨⟨T, fun a b ↦ ?_, fun a b ↦ ?_⟩, rfl⟩
· exact congr($(h (L a) (.inl ⟨a, rfl⟩)) b).symm
· exact congr($(h (R b) (.inr ⟨b, rfl⟩)) a).symm
/-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z :=
{ L (z : α) with
map_mul_left' := ((Set.mem_center_iff _).mp z.prop).left_comm
map_mul_right' := ((Set.mem_center_iff _).mp z.prop).left_assoc }
map_zero' := by
simp only [ZeroMemClass.coe_zero, map_zero]
exact rfl
map_add' := fun _ _ => by
simp only [AddSubmonoid.coe_add, NonUnitalSubsemiring.coe_toAddSubmonoid, map_add]
exact rfl
map_mul' := fun z₁ z₂ => by
ext a
exact (((Set.mem_center_iff _).mp z₁.prop).left_assoc z₂ a).symm
lemma centerToCentroid_apply (z : { x // x ∈ NonUnitalSubsemiring.center α }) (a : α) :
(centerToCentroid z) a = z * a := rfl
lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
have e2 (d : α) : T d = d * a := congr($(hT.trans hc) d)
constructor
case comm => exact (congr($hc ·))
case left_assoc => simpa [e1] using (map_mul_right T · ·)
case mid_assoc => exact fun b c ↦ by simpa [e1 c, e2 b] using
(map_mul_right T b c).symm.trans <| map_mul_left T b c
case right_assoc => simpa [e2] using (map_mul_left T · ·)
end NonUnitalNonAssocSemiring
section NonAssocSemiring
variable [NonAssocSemiring α]
/-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by refine ⟨?_, ?_, ?_, ?_⟩; all_goals simp [← map_mul_left, ← map_mul_right]⟩
left_inv := fun z ↦ Subtype.ext <| by simp [centerToCentroid_apply]
right_inv := fun T ↦ CentroidHom.ext <| by simp [centerToCentroid_apply, ← map_mul_right] }
end NonAssocSemiring
section NonUnitalNonAssocRing
variable [NonUnitalNonAssocRing α]
/-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) :=
⟨fun f ↦
{ (-f : α →+ α) with
map_mul_left' := fun a b ↦ by
change -f (a * b) = a * (-f b)
simp [map_mul_left]
map_mul_right' := fun a b ↦ by
change -f (a * b) = (-f a) * b
simp [map_mul_right] }⟩
instance : Sub (CentroidHom α) :=
⟨fun f g ↦
{ (f - g : α →+ α) with
map_mul_left' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = a * (⇑f - ⇑g) b
simp [map_mul_left, mul_sub]
map_mul_right' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = ((⇑f - ⇑g) a) * b
simp [map_mul_right, sub_mul] }⟩
#noalign centroid_hom.has_zsmul
instance : IntCast (CentroidHom α) where intCast z := z • (1 : CentroidHom α)
-- Porting note: `nolint simpNF` added because simplify fails on left-hand side
@[simp, norm_cast, nolint simpNF]
theorem coe_int_cast (z : ℤ) : ⇑(z : CentroidHom α) = z • (CentroidHom.id α) :=
rfl
#align centroid_hom.coe_int_cast CentroidHom.coe_int_cast
theorem int_cast_apply (z : ℤ) (m : α) : (z : CentroidHom α) m = z • m :=
rfl
#align centroid_hom.int_cast_apply CentroidHom.int_cast_apply
@[simp]
theorem toEnd_neg (x : CentroidHom α) : (-x).toEnd = -x.toEnd :=
rfl
#align centroid_hom.to_End_neg CentroidHom.toEnd_neg
@[simp]
theorem toEnd_sub (x y : CentroidHom α) : (x - y).toEnd = x.toEnd - y.toEnd :=
rfl
#align centroid_hom.to_End_sub CentroidHom.toEnd_sub
#align centroid_hom.to_End_zsmul CentroidHom.toEnd_smul
instance : AddCommGroup (CentroidHom α) :=
toEnd_injective.addCommGroup _
toEnd_zero toEnd_add toEnd_neg toEnd_sub (swap toEnd_smul) (swap toEnd_smul)
@[simp, norm_cast]
theorem coe_neg (f : CentroidHom α) : ⇑(-f) = -f :=
rfl
#align centroid_hom.coe_neg CentroidHom.coe_neg
@[simp, norm_cast]
theorem coe_sub (f g : CentroidHom α) : ⇑(f - g) = f - g :=
rfl
#align centroid_hom.coe_sub CentroidHom.coe_sub
@[simp]
theorem neg_apply (f : CentroidHom α) (a : α) : (-f) a = -f a :=
rfl
#align centroid_hom.neg_apply CentroidHom.neg_apply
@[simp]
theorem sub_apply (f g : CentroidHom α) (a : α) : (f - g) a = f a - g a :=
rfl
#align centroid_hom.sub_apply CentroidHom.sub_apply
@[simp, norm_cast]
theorem toEnd_int_cast (z : ℤ) : (z : CentroidHom α).toEnd = ↑z :=
rfl
#align centroid_hom.to_End_int_cast CentroidHom.toEnd_int_cast
instance instRing : Ring (CentroidHom α) :=
toEnd_injective.ring _ toEnd_zero toEnd_one toEnd_add toEnd_mul toEnd_neg toEnd_sub
(swap toEnd_smul) (swap toEnd_smul) toEnd_pow toEnd_nat_cast toEnd_int_cast
end NonUnitalNonAssocRing
section NonUnitalRing
variable [NonUnitalRing α]
-- Porting note: Not sure why Lean didn't like `CentroidHom.Ring`
-- See note [reducible non instances]
/-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) :=
{ CentroidHom.instRing with
mul_comm := fun f g ↦ by
| ext | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) :=
{ CentroidHom.instRing with
mul_comm := fun f g ↦ by
| Mathlib.Algebra.Ring.CentroidHom.591_0.FQQ3LT1tg3cKlkH | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
case h
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalRing α
h : ∀ (a b : α), (∀ (r : α), a * r * b = 0) → a = 0 ∨ b = 0
src✝ : Ring (CentroidHom α) := instRing
f g : CentroidHom α
a✝ : α
⊢ (f * g) a✝ = (g * f) a✝ | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import algebra.hom.centroid from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3ad319514ff"
/-!
# Centroid homomorphisms
Let `A` be a (non unital, non associative) algebra. The centroid of `A` is the set of linear maps
`T` on `A` such that `T` commutes with left and right multiplication, that is to say, for all `a`
and `b` in `A`,
$$
T(ab) = (Ta)b, T(ab) = a(Tb).
$$
In mathlib we call elements of the centroid "centroid homomorphisms" (`CentroidHom`) in keeping
with `AddMonoidHom` etc.
We use the `FunLike` design, so each type of morphisms has a companion typeclass which is meant to
be satisfied by itself and all stricter types.
## Types of morphisms
* `CentroidHom`: Maps which preserve left and right multiplication.
## Typeclasses
* `CentroidHomClass`
## References
* [Jacobson, Structure of Rings][Jacobson1956]
* [McCrimmon, A taste of Jordan algebras][mccrimmon2004]
## Tags
centroid
-/
open Function
variable {F M N R α : Type*}
/-- The type of centroid homomorphisms from `α` to `α`. -/
structure CentroidHom (α : Type*) [NonUnitalNonAssocSemiring α] extends α →+ α where
/-- Commutativity of centroid homomorphims with left multiplication. -/
map_mul_left' (a b : α) : toFun (a * b) = a * toFun b
/-- Commutativity of centroid homomorphims with right multiplication. -/
map_mul_right' (a b : α) : toFun (a * b) = toFun a * b
#align centroid_hom CentroidHom
attribute [nolint docBlame] CentroidHom.toAddMonoidHom
/-- `CentroidHomClass F α` states that `F` is a type of centroid homomorphisms.
You should extend this class when you extend `CentroidHom`. -/
class CentroidHomClass (F : Type*) (α : outParam <| Type*) [NonUnitalNonAssocSemiring α] extends
AddMonoidHomClass F α α where
/-- Commutativity of centroid homomorphims with left multiplication. -/
map_mul_left (f : F) (a b : α) : f (a * b) = a * f b
/-- Commutativity of centroid homomorphims with right multiplication. -/
map_mul_right (f : F) (a b : α) : f (a * b) = f a * b
#align centroid_hom_class CentroidHomClass
export CentroidHomClass (map_mul_left map_mul_right)
instance [NonUnitalNonAssocSemiring α] [CentroidHomClass F α] : CoeTC F (CentroidHom α) :=
⟨fun f ↦
{ (f : α →+ α) with
toFun := f
map_mul_left' := map_mul_left f
map_mul_right' := map_mul_right f }⟩
/-! ### Centroid homomorphisms -/
namespace CentroidHom
section NonUnitalNonAssocSemiring
variable [NonUnitalNonAssocSemiring α]
instance : CentroidHomClass (CentroidHom α) α where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
congr with x
exact congrFun h x
map_zero f := f.map_zero'
map_add f := f.map_add'
map_mul_left f := f.map_mul_left'
map_mul_right f := f.map_mul_right'
/-- Helper instance for when there's too many metavariables to apply `FunLike.CoeFun`
directly. -/
/- Porting note: Lean gave me `unknown constant 'FunLike.CoeFun'` and says `CoeFun` is a type
mismatch, so I used `library_search`. -/
instance : CoeFun (CentroidHom α) fun _ ↦ α → α :=
inferInstanceAs (CoeFun (CentroidHom α) fun _ ↦ α → α)
-- Porting note: removed @[simp]; not in normal form. (`toAddMonoidHom_eq_coe` below ensures that
-- the LHS simplifies to the RHS anyway.)
theorem toFun_eq_coe {f : CentroidHom α} : f.toFun = f := rfl
#align centroid_hom.to_fun_eq_coe CentroidHom.toFun_eq_coe
@[ext]
theorem ext {f g : CentroidHom α} (h : ∀ a, f a = g a) : f = g :=
FunLike.ext f g h
#align centroid_hom.ext CentroidHom.ext
@[simp, norm_cast]
theorem coe_toAddMonoidHom (f : CentroidHom α) : ⇑(f : α →+ α) = f :=
rfl
#align centroid_hom.coe_to_add_monoid_hom CentroidHom.coe_toAddMonoidHom
@[simp]
theorem toAddMonoidHom_eq_coe (f : CentroidHom α) : f.toAddMonoidHom = f :=
rfl
#align centroid_hom.to_add_monoid_hom_eq_coe CentroidHom.toAddMonoidHom_eq_coe
theorem coe_toAddMonoidHom_injective : Injective ((↑) : CentroidHom α → α →+ α) :=
fun _f _g h => ext fun a ↦
haveI := FunLike.congr_fun h a
this
#align centroid_hom.coe_to_add_monoid_hom_injective CentroidHom.coe_toAddMonoidHom_injective
/-- Turn a centroid homomorphism into an additive monoid endomorphism. -/
def toEnd (f : CentroidHom α) : AddMonoid.End α :=
(f : α →+ α)
#align centroid_hom.to_End CentroidHom.toEnd
theorem toEnd_injective : Injective (CentroidHom.toEnd : CentroidHom α → AddMonoid.End α) :=
coe_toAddMonoidHom_injective
#align centroid_hom.to_End_injective CentroidHom.toEnd_injective
/-- Copy of a `CentroidHom` with a new `toFun` equal to the old one. Useful to fix
definitional equalities. -/
protected def copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : CentroidHom α :=
{ f.toAddMonoidHom.copy f' <| h with
toFun := f'
map_mul_left' := fun a b ↦ by simp_rw [h, map_mul_left]
map_mul_right' := fun a b ↦ by simp_rw [h, map_mul_right] }
#align centroid_hom.copy CentroidHom.copy
@[simp]
theorem coe_copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : ⇑(f.copy f' h) = f' :=
rfl
#align centroid_hom.coe_copy CentroidHom.coe_copy
theorem copy_eq (f : CentroidHom α) (f' : α → α) (h : f' = f) : f.copy f' h = f :=
FunLike.ext' h
#align centroid_hom.copy_eq CentroidHom.copy_eq
variable (α)
/-- `id` as a `CentroidHom`. -/
protected def id : CentroidHom α :=
{ AddMonoidHom.id α with
map_mul_left' := fun _ _ ↦ rfl
map_mul_right' := fun _ _ ↦ rfl }
#align centroid_hom.id CentroidHom.id
instance : Inhabited (CentroidHom α) :=
⟨CentroidHom.id α⟩
@[simp, norm_cast]
theorem coe_id : ⇑(CentroidHom.id α) = id :=
rfl
#align centroid_hom.coe_id CentroidHom.coe_id
@[simp, norm_cast]
theorem toAddMonoidHom_id : (CentroidHom.id α : α →+ α) = AddMonoidHom.id α :=
rfl
#align centroid_hom.coe_to_add_monoid_hom_id CentroidHom.toAddMonoidHom_id
variable {α}
@[simp]
theorem id_apply (a : α) : CentroidHom.id α a = a :=
rfl
#align centroid_hom.id_apply CentroidHom.id_apply
/-- Composition of `CentroidHom`s as a `CentroidHom`. -/
def comp (g f : CentroidHom α) : CentroidHom α :=
{ g.toAddMonoidHom.comp f.toAddMonoidHom with
map_mul_left' := fun _a _b ↦ (congr_arg g <| f.map_mul_left' _ _).trans <| g.map_mul_left' _ _
map_mul_right' := fun _a _b ↦
(congr_arg g <| f.map_mul_right' _ _).trans <| g.map_mul_right' _ _ }
#align centroid_hom.comp CentroidHom.comp
@[simp, norm_cast]
theorem coe_comp (g f : CentroidHom α) : ⇑(g.comp f) = g ∘ f :=
rfl
#align centroid_hom.coe_comp CentroidHom.coe_comp
@[simp]
theorem comp_apply (g f : CentroidHom α) (a : α) : g.comp f a = g (f a) :=
rfl
#align centroid_hom.comp_apply CentroidHom.comp_apply
@[simp, norm_cast]
theorem coe_comp_addMonoidHom (g f : CentroidHom α) : (g.comp f : α →+ α) = (g : α →+ α).comp f :=
rfl
#align centroid_hom.coe_comp_add_monoid_hom CentroidHom.coe_comp_addMonoidHom
@[simp]
theorem comp_assoc (h g f : CentroidHom α) : (h.comp g).comp f = h.comp (g.comp f) :=
rfl
#align centroid_hom.comp_assoc CentroidHom.comp_assoc
@[simp]
theorem comp_id (f : CentroidHom α) : f.comp (CentroidHom.id α) = f :=
rfl
#align centroid_hom.comp_id CentroidHom.comp_id
@[simp]
theorem id_comp (f : CentroidHom α) : (CentroidHom.id α).comp f = f :=
rfl
#align centroid_hom.id_comp CentroidHom.id_comp
@[simp]
theorem cancel_right {g₁ g₂ f : CentroidHom α} (hf : Surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨fun h ↦ ext <| hf.forall.2 <| FunLike.ext_iff.1 h, fun a ↦ congrFun (congrArg comp a) f⟩
#align centroid_hom.cancel_right CentroidHom.cancel_right
@[simp]
theorem cancel_left {g f₁ f₂ : CentroidHom α} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨fun h ↦ ext fun a ↦ hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩
#align centroid_hom.cancel_left CentroidHom.cancel_left
instance : Zero (CentroidHom α) :=
⟨{ (0 : α →+ α) with
map_mul_left' := fun _a _b ↦ (mul_zero _).symm
map_mul_right' := fun _a _b ↦ (zero_mul _).symm }⟩
instance : One (CentroidHom α) :=
⟨CentroidHom.id α⟩
instance : Add (CentroidHom α) :=
⟨fun f g ↦
{ (f + g : α →+ α) with
map_mul_left' := fun a b ↦ by
show f (a * b) + g (a * b) = a * (f b + g b)
simp [map_mul_left, mul_add]
map_mul_right' := fun a b ↦ by
show f (a * b) + g (a * b) = (f a + g a) * b
simp [map_mul_right, add_mul] }⟩
instance : Mul (CentroidHom α) :=
⟨comp⟩
variable [Monoid M] [Monoid N] [Semiring R]
variable [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α]
variable [DistribMulAction N α] [SMulCommClass N α α] [IsScalarTower N α α]
variable [Module R α] [SMulCommClass R α α] [IsScalarTower R α α]
instance instSMul : SMul M (CentroidHom α) where
smul n f :=
{ (n • f : α →+ α) with
map_mul_left' := fun a b ↦ by
change n • f (a * b) = a * n • f b
rw [map_mul_left f, ← mul_smul_comm]
map_mul_right' := fun a b ↦ by
change n • f (a * b) = n • f a * b
rw [map_mul_right f, ← smul_mul_assoc] }
#noalign centroid_hom.has_nsmul
instance [SMul M N] [IsScalarTower M N α] : IsScalarTower M N (CentroidHom α) where
smul_assoc _ _ _ := ext <| fun _ => smul_assoc _ _ _
instance [SMulCommClass M N α] : SMulCommClass M N (CentroidHom α) where
smul_comm _ _ _ := ext <| fun _ => smul_comm _ _ _
instance [DistribMulAction Mᵐᵒᵖ α] [IsCentralScalar M α] : IsCentralScalar M (CentroidHom α) where
op_smul_eq_smul _ _ := ext <| fun _ => op_smul_eq_smul _ _
instance isScalarTowerRight : IsScalarTower M (CentroidHom α) (CentroidHom α) where
smul_assoc _ _ _ := rfl
instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_left' _ _)
map_mul_right' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_right' _ _) }⟩
#align centroid_hom.has_npow_nat CentroidHom.hasNPowNat
@[simp, norm_cast]
theorem coe_zero : ⇑(0 : CentroidHom α) = 0 :=
rfl
#align centroid_hom.coe_zero CentroidHom.coe_zero
@[simp, norm_cast]
theorem coe_one : ⇑(1 : CentroidHom α) = id :=
rfl
#align centroid_hom.coe_one CentroidHom.coe_one
@[simp, norm_cast]
theorem coe_add (f g : CentroidHom α) : ⇑(f + g) = f + g :=
rfl
#align centroid_hom.coe_add CentroidHom.coe_add
@[simp, norm_cast]
theorem coe_mul (f g : CentroidHom α) : ⇑(f * g) = f ∘ g :=
rfl
#align centroid_hom.coe_mul CentroidHom.coe_mul
@[simp, norm_cast]
theorem coe_smul (n : M) (f : CentroidHom α) : ⇑(n • f) = n • ⇑f :=
rfl
#align centroid_hom.coe_nsmul CentroidHom.coe_smul
@[simp]
theorem zero_apply (a : α) : (0 : CentroidHom α) a = 0 :=
rfl
#align centroid_hom.zero_apply CentroidHom.zero_apply
@[simp]
theorem one_apply (a : α) : (1 : CentroidHom α) a = a :=
rfl
#align centroid_hom.one_apply CentroidHom.one_apply
@[simp]
theorem add_apply (f g : CentroidHom α) (a : α) : (f + g) a = f a + g a :=
rfl
#align centroid_hom.add_apply CentroidHom.add_apply
@[simp]
theorem mul_apply (f g : CentroidHom α) (a : α) : (f * g) a = f (g a) :=
rfl
#align centroid_hom.mul_apply CentroidHom.mul_apply
@[simp]
theorem smul_apply (n : M) (f : CentroidHom α) (a : α) : (n • f) a = n • f a :=
rfl
#align centroid_hom.nsmul_apply CentroidHom.smul_apply
example : SMul ℕ (CentroidHom α) := instSMul
@[simp]
theorem toEnd_zero : (0 : CentroidHom α).toEnd = 0 :=
rfl
#align centroid_hom.to_End_zero CentroidHom.toEnd_zero
@[simp]
theorem toEnd_add (x y : CentroidHom α) : (x + y).toEnd = x.toEnd + y.toEnd :=
rfl
#align centroid_hom.to_End_add CentroidHom.toEnd_add
theorem toEnd_smul (m : M) (x : CentroidHom α) : (m • x).toEnd = m • x.toEnd :=
rfl
#align centroid_hom.to_End_nsmul CentroidHom.toEnd_smul
instance : AddCommMonoid (CentroidHom α) :=
coe_toAddMonoidHom_injective.addCommMonoid _ toEnd_zero toEnd_add (swap toEnd_smul)
instance : NatCast (CentroidHom α) where natCast n := n • (1 : CentroidHom α)
-- Porting note: `nolint simpNF` added because simplify fails on left-hand side
@[simp, norm_cast, nolint simpNF]
theorem coe_nat_cast (n : ℕ) : ⇑(n : CentroidHom α) = n • (CentroidHom.id α) :=
rfl
#align centroid_hom.coe_nat_cast CentroidHom.coe_nat_cast
theorem nat_cast_apply (n : ℕ) (m : α) : (n : CentroidHom α) m = n • m :=
rfl
#align centroid_hom.nat_cast_apply CentroidHom.nat_cast_apply
@[simp]
theorem toEnd_one : (1 : CentroidHom α).toEnd = 1 :=
rfl
#align centroid_hom.to_End_one CentroidHom.toEnd_one
@[simp]
theorem toEnd_mul (x y : CentroidHom α) : (x * y).toEnd = x.toEnd * y.toEnd :=
rfl
#align centroid_hom.to_End_mul CentroidHom.toEnd_mul
@[simp]
theorem toEnd_pow (x : CentroidHom α) (n : ℕ) : (x ^ n).toEnd = x.toEnd ^ n :=
rfl
#align centroid_hom.to_End_pow CentroidHom.toEnd_pow
@[simp, norm_cast]
theorem toEnd_nat_cast (n : ℕ) : (n : CentroidHom α).toEnd = ↑n :=
rfl
#align centroid_hom.to_End_nat_cast CentroidHom.toEnd_nat_cast
-- cf `add_monoid.End.semiring`
instance : Semiring (CentroidHom α) :=
toEnd_injective.semiring _ toEnd_zero toEnd_one toEnd_add toEnd_mul (swap toEnd_smul) toEnd_pow
toEnd_nat_cast
variable (α) in
/-- `CentroidHom.toEnd` as a `RingHom`. -/
@[simps]
def toEndRingHom : CentroidHom α →+* AddMonoid.End α where
toFun := toEnd
map_zero' := toEnd_zero
map_one' := toEnd_one
map_add' := toEnd_add
map_mul' := toEnd_mul
theorem comp_mul_comm (T S : CentroidHom α) (a b : α) : (T ∘ S) (a * b) = (S ∘ T) (a * b) := by
simp only [Function.comp_apply]
rw [map_mul_right, map_mul_left, ← map_mul_right, ← map_mul_left]
#align centroid_hom.comp_mul_comm CentroidHom.comp_mul_comm
instance : DistribMulAction M (CentroidHom α) :=
toEnd_injective.distribMulAction (toEndRingHom α).toAddMonoidHom toEnd_smul
instance : Module R (CentroidHom α) :=
toEnd_injective.module R (toEndRingHom α).toAddMonoidHom toEnd_smul
local notation "L" => AddMonoid.End.mulLeft
local notation "R" => AddMonoid.End.mulRight
lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
refine ⟨?_, fun h ↦ ?_⟩
· rintro ⟨f, rfl⟩ S (⟨a, rfl⟩ | ⟨b, rfl⟩)
· exact AddMonoidHom.ext fun b ↦ (map_mul_left f a b).symm
· exact AddMonoidHom.ext fun a ↦ (map_mul_right f a b).symm
· rw [Subsemiring.mem_centralizer_iff] at h
refine ⟨⟨T, fun a b ↦ ?_, fun a b ↦ ?_⟩, rfl⟩
· exact congr($(h (L a) (.inl ⟨a, rfl⟩)) b).symm
· exact congr($(h (R b) (.inr ⟨b, rfl⟩)) a).symm
/-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z :=
{ L (z : α) with
map_mul_left' := ((Set.mem_center_iff _).mp z.prop).left_comm
map_mul_right' := ((Set.mem_center_iff _).mp z.prop).left_assoc }
map_zero' := by
simp only [ZeroMemClass.coe_zero, map_zero]
exact rfl
map_add' := fun _ _ => by
simp only [AddSubmonoid.coe_add, NonUnitalSubsemiring.coe_toAddSubmonoid, map_add]
exact rfl
map_mul' := fun z₁ z₂ => by
ext a
exact (((Set.mem_center_iff _).mp z₁.prop).left_assoc z₂ a).symm
lemma centerToCentroid_apply (z : { x // x ∈ NonUnitalSubsemiring.center α }) (a : α) :
(centerToCentroid z) a = z * a := rfl
lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
have e2 (d : α) : T d = d * a := congr($(hT.trans hc) d)
constructor
case comm => exact (congr($hc ·))
case left_assoc => simpa [e1] using (map_mul_right T · ·)
case mid_assoc => exact fun b c ↦ by simpa [e1 c, e2 b] using
(map_mul_right T b c).symm.trans <| map_mul_left T b c
case right_assoc => simpa [e2] using (map_mul_left T · ·)
end NonUnitalNonAssocSemiring
section NonAssocSemiring
variable [NonAssocSemiring α]
/-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by refine ⟨?_, ?_, ?_, ?_⟩; all_goals simp [← map_mul_left, ← map_mul_right]⟩
left_inv := fun z ↦ Subtype.ext <| by simp [centerToCentroid_apply]
right_inv := fun T ↦ CentroidHom.ext <| by simp [centerToCentroid_apply, ← map_mul_right] }
end NonAssocSemiring
section NonUnitalNonAssocRing
variable [NonUnitalNonAssocRing α]
/-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) :=
⟨fun f ↦
{ (-f : α →+ α) with
map_mul_left' := fun a b ↦ by
change -f (a * b) = a * (-f b)
simp [map_mul_left]
map_mul_right' := fun a b ↦ by
change -f (a * b) = (-f a) * b
simp [map_mul_right] }⟩
instance : Sub (CentroidHom α) :=
⟨fun f g ↦
{ (f - g : α →+ α) with
map_mul_left' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = a * (⇑f - ⇑g) b
simp [map_mul_left, mul_sub]
map_mul_right' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = ((⇑f - ⇑g) a) * b
simp [map_mul_right, sub_mul] }⟩
#noalign centroid_hom.has_zsmul
instance : IntCast (CentroidHom α) where intCast z := z • (1 : CentroidHom α)
-- Porting note: `nolint simpNF` added because simplify fails on left-hand side
@[simp, norm_cast, nolint simpNF]
theorem coe_int_cast (z : ℤ) : ⇑(z : CentroidHom α) = z • (CentroidHom.id α) :=
rfl
#align centroid_hom.coe_int_cast CentroidHom.coe_int_cast
theorem int_cast_apply (z : ℤ) (m : α) : (z : CentroidHom α) m = z • m :=
rfl
#align centroid_hom.int_cast_apply CentroidHom.int_cast_apply
@[simp]
theorem toEnd_neg (x : CentroidHom α) : (-x).toEnd = -x.toEnd :=
rfl
#align centroid_hom.to_End_neg CentroidHom.toEnd_neg
@[simp]
theorem toEnd_sub (x y : CentroidHom α) : (x - y).toEnd = x.toEnd - y.toEnd :=
rfl
#align centroid_hom.to_End_sub CentroidHom.toEnd_sub
#align centroid_hom.to_End_zsmul CentroidHom.toEnd_smul
instance : AddCommGroup (CentroidHom α) :=
toEnd_injective.addCommGroup _
toEnd_zero toEnd_add toEnd_neg toEnd_sub (swap toEnd_smul) (swap toEnd_smul)
@[simp, norm_cast]
theorem coe_neg (f : CentroidHom α) : ⇑(-f) = -f :=
rfl
#align centroid_hom.coe_neg CentroidHom.coe_neg
@[simp, norm_cast]
theorem coe_sub (f g : CentroidHom α) : ⇑(f - g) = f - g :=
rfl
#align centroid_hom.coe_sub CentroidHom.coe_sub
@[simp]
theorem neg_apply (f : CentroidHom α) (a : α) : (-f) a = -f a :=
rfl
#align centroid_hom.neg_apply CentroidHom.neg_apply
@[simp]
theorem sub_apply (f g : CentroidHom α) (a : α) : (f - g) a = f a - g a :=
rfl
#align centroid_hom.sub_apply CentroidHom.sub_apply
@[simp, norm_cast]
theorem toEnd_int_cast (z : ℤ) : (z : CentroidHom α).toEnd = ↑z :=
rfl
#align centroid_hom.to_End_int_cast CentroidHom.toEnd_int_cast
instance instRing : Ring (CentroidHom α) :=
toEnd_injective.ring _ toEnd_zero toEnd_one toEnd_add toEnd_mul toEnd_neg toEnd_sub
(swap toEnd_smul) (swap toEnd_smul) toEnd_pow toEnd_nat_cast toEnd_int_cast
end NonUnitalNonAssocRing
section NonUnitalRing
variable [NonUnitalRing α]
-- Porting note: Not sure why Lean didn't like `CentroidHom.Ring`
-- See note [reducible non instances]
/-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) :=
{ CentroidHom.instRing with
mul_comm := fun f g ↦ by
ext
| refine' sub_eq_zero.1 (or_self_iff.1 <| (h _ _) fun r ↦ _) | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) :=
{ CentroidHom.instRing with
mul_comm := fun f g ↦ by
ext
| Mathlib.Algebra.Ring.CentroidHom.591_0.FQQ3LT1tg3cKlkH | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
case h
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalRing α
h : ∀ (a b : α), (∀ (r : α), a * r * b = 0) → a = 0 ∨ b = 0
src✝ : Ring (CentroidHom α) := instRing
f g : CentroidHom α
a✝ r : α
⊢ ((f * g) a✝ - (g * f) a✝) * r * ((f * g) a✝ - (g * f) a✝) = 0 | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import algebra.hom.centroid from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3ad319514ff"
/-!
# Centroid homomorphisms
Let `A` be a (non unital, non associative) algebra. The centroid of `A` is the set of linear maps
`T` on `A` such that `T` commutes with left and right multiplication, that is to say, for all `a`
and `b` in `A`,
$$
T(ab) = (Ta)b, T(ab) = a(Tb).
$$
In mathlib we call elements of the centroid "centroid homomorphisms" (`CentroidHom`) in keeping
with `AddMonoidHom` etc.
We use the `FunLike` design, so each type of morphisms has a companion typeclass which is meant to
be satisfied by itself and all stricter types.
## Types of morphisms
* `CentroidHom`: Maps which preserve left and right multiplication.
## Typeclasses
* `CentroidHomClass`
## References
* [Jacobson, Structure of Rings][Jacobson1956]
* [McCrimmon, A taste of Jordan algebras][mccrimmon2004]
## Tags
centroid
-/
open Function
variable {F M N R α : Type*}
/-- The type of centroid homomorphisms from `α` to `α`. -/
structure CentroidHom (α : Type*) [NonUnitalNonAssocSemiring α] extends α →+ α where
/-- Commutativity of centroid homomorphims with left multiplication. -/
map_mul_left' (a b : α) : toFun (a * b) = a * toFun b
/-- Commutativity of centroid homomorphims with right multiplication. -/
map_mul_right' (a b : α) : toFun (a * b) = toFun a * b
#align centroid_hom CentroidHom
attribute [nolint docBlame] CentroidHom.toAddMonoidHom
/-- `CentroidHomClass F α` states that `F` is a type of centroid homomorphisms.
You should extend this class when you extend `CentroidHom`. -/
class CentroidHomClass (F : Type*) (α : outParam <| Type*) [NonUnitalNonAssocSemiring α] extends
AddMonoidHomClass F α α where
/-- Commutativity of centroid homomorphims with left multiplication. -/
map_mul_left (f : F) (a b : α) : f (a * b) = a * f b
/-- Commutativity of centroid homomorphims with right multiplication. -/
map_mul_right (f : F) (a b : α) : f (a * b) = f a * b
#align centroid_hom_class CentroidHomClass
export CentroidHomClass (map_mul_left map_mul_right)
instance [NonUnitalNonAssocSemiring α] [CentroidHomClass F α] : CoeTC F (CentroidHom α) :=
⟨fun f ↦
{ (f : α →+ α) with
toFun := f
map_mul_left' := map_mul_left f
map_mul_right' := map_mul_right f }⟩
/-! ### Centroid homomorphisms -/
namespace CentroidHom
section NonUnitalNonAssocSemiring
variable [NonUnitalNonAssocSemiring α]
instance : CentroidHomClass (CentroidHom α) α where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
congr with x
exact congrFun h x
map_zero f := f.map_zero'
map_add f := f.map_add'
map_mul_left f := f.map_mul_left'
map_mul_right f := f.map_mul_right'
/-- Helper instance for when there's too many metavariables to apply `FunLike.CoeFun`
directly. -/
/- Porting note: Lean gave me `unknown constant 'FunLike.CoeFun'` and says `CoeFun` is a type
mismatch, so I used `library_search`. -/
instance : CoeFun (CentroidHom α) fun _ ↦ α → α :=
inferInstanceAs (CoeFun (CentroidHom α) fun _ ↦ α → α)
-- Porting note: removed @[simp]; not in normal form. (`toAddMonoidHom_eq_coe` below ensures that
-- the LHS simplifies to the RHS anyway.)
theorem toFun_eq_coe {f : CentroidHom α} : f.toFun = f := rfl
#align centroid_hom.to_fun_eq_coe CentroidHom.toFun_eq_coe
@[ext]
theorem ext {f g : CentroidHom α} (h : ∀ a, f a = g a) : f = g :=
FunLike.ext f g h
#align centroid_hom.ext CentroidHom.ext
@[simp, norm_cast]
theorem coe_toAddMonoidHom (f : CentroidHom α) : ⇑(f : α →+ α) = f :=
rfl
#align centroid_hom.coe_to_add_monoid_hom CentroidHom.coe_toAddMonoidHom
@[simp]
theorem toAddMonoidHom_eq_coe (f : CentroidHom α) : f.toAddMonoidHom = f :=
rfl
#align centroid_hom.to_add_monoid_hom_eq_coe CentroidHom.toAddMonoidHom_eq_coe
theorem coe_toAddMonoidHom_injective : Injective ((↑) : CentroidHom α → α →+ α) :=
fun _f _g h => ext fun a ↦
haveI := FunLike.congr_fun h a
this
#align centroid_hom.coe_to_add_monoid_hom_injective CentroidHom.coe_toAddMonoidHom_injective
/-- Turn a centroid homomorphism into an additive monoid endomorphism. -/
def toEnd (f : CentroidHom α) : AddMonoid.End α :=
(f : α →+ α)
#align centroid_hom.to_End CentroidHom.toEnd
theorem toEnd_injective : Injective (CentroidHom.toEnd : CentroidHom α → AddMonoid.End α) :=
coe_toAddMonoidHom_injective
#align centroid_hom.to_End_injective CentroidHom.toEnd_injective
/-- Copy of a `CentroidHom` with a new `toFun` equal to the old one. Useful to fix
definitional equalities. -/
protected def copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : CentroidHom α :=
{ f.toAddMonoidHom.copy f' <| h with
toFun := f'
map_mul_left' := fun a b ↦ by simp_rw [h, map_mul_left]
map_mul_right' := fun a b ↦ by simp_rw [h, map_mul_right] }
#align centroid_hom.copy CentroidHom.copy
@[simp]
theorem coe_copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : ⇑(f.copy f' h) = f' :=
rfl
#align centroid_hom.coe_copy CentroidHom.coe_copy
theorem copy_eq (f : CentroidHom α) (f' : α → α) (h : f' = f) : f.copy f' h = f :=
FunLike.ext' h
#align centroid_hom.copy_eq CentroidHom.copy_eq
variable (α)
/-- `id` as a `CentroidHom`. -/
protected def id : CentroidHom α :=
{ AddMonoidHom.id α with
map_mul_left' := fun _ _ ↦ rfl
map_mul_right' := fun _ _ ↦ rfl }
#align centroid_hom.id CentroidHom.id
instance : Inhabited (CentroidHom α) :=
⟨CentroidHom.id α⟩
@[simp, norm_cast]
theorem coe_id : ⇑(CentroidHom.id α) = id :=
rfl
#align centroid_hom.coe_id CentroidHom.coe_id
@[simp, norm_cast]
theorem toAddMonoidHom_id : (CentroidHom.id α : α →+ α) = AddMonoidHom.id α :=
rfl
#align centroid_hom.coe_to_add_monoid_hom_id CentroidHom.toAddMonoidHom_id
variable {α}
@[simp]
theorem id_apply (a : α) : CentroidHom.id α a = a :=
rfl
#align centroid_hom.id_apply CentroidHom.id_apply
/-- Composition of `CentroidHom`s as a `CentroidHom`. -/
def comp (g f : CentroidHom α) : CentroidHom α :=
{ g.toAddMonoidHom.comp f.toAddMonoidHom with
map_mul_left' := fun _a _b ↦ (congr_arg g <| f.map_mul_left' _ _).trans <| g.map_mul_left' _ _
map_mul_right' := fun _a _b ↦
(congr_arg g <| f.map_mul_right' _ _).trans <| g.map_mul_right' _ _ }
#align centroid_hom.comp CentroidHom.comp
@[simp, norm_cast]
theorem coe_comp (g f : CentroidHom α) : ⇑(g.comp f) = g ∘ f :=
rfl
#align centroid_hom.coe_comp CentroidHom.coe_comp
@[simp]
theorem comp_apply (g f : CentroidHom α) (a : α) : g.comp f a = g (f a) :=
rfl
#align centroid_hom.comp_apply CentroidHom.comp_apply
@[simp, norm_cast]
theorem coe_comp_addMonoidHom (g f : CentroidHom α) : (g.comp f : α →+ α) = (g : α →+ α).comp f :=
rfl
#align centroid_hom.coe_comp_add_monoid_hom CentroidHom.coe_comp_addMonoidHom
@[simp]
theorem comp_assoc (h g f : CentroidHom α) : (h.comp g).comp f = h.comp (g.comp f) :=
rfl
#align centroid_hom.comp_assoc CentroidHom.comp_assoc
@[simp]
theorem comp_id (f : CentroidHom α) : f.comp (CentroidHom.id α) = f :=
rfl
#align centroid_hom.comp_id CentroidHom.comp_id
@[simp]
theorem id_comp (f : CentroidHom α) : (CentroidHom.id α).comp f = f :=
rfl
#align centroid_hom.id_comp CentroidHom.id_comp
@[simp]
theorem cancel_right {g₁ g₂ f : CentroidHom α} (hf : Surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨fun h ↦ ext <| hf.forall.2 <| FunLike.ext_iff.1 h, fun a ↦ congrFun (congrArg comp a) f⟩
#align centroid_hom.cancel_right CentroidHom.cancel_right
@[simp]
theorem cancel_left {g f₁ f₂ : CentroidHom α} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨fun h ↦ ext fun a ↦ hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩
#align centroid_hom.cancel_left CentroidHom.cancel_left
instance : Zero (CentroidHom α) :=
⟨{ (0 : α →+ α) with
map_mul_left' := fun _a _b ↦ (mul_zero _).symm
map_mul_right' := fun _a _b ↦ (zero_mul _).symm }⟩
instance : One (CentroidHom α) :=
⟨CentroidHom.id α⟩
instance : Add (CentroidHom α) :=
⟨fun f g ↦
{ (f + g : α →+ α) with
map_mul_left' := fun a b ↦ by
show f (a * b) + g (a * b) = a * (f b + g b)
simp [map_mul_left, mul_add]
map_mul_right' := fun a b ↦ by
show f (a * b) + g (a * b) = (f a + g a) * b
simp [map_mul_right, add_mul] }⟩
instance : Mul (CentroidHom α) :=
⟨comp⟩
variable [Monoid M] [Monoid N] [Semiring R]
variable [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α]
variable [DistribMulAction N α] [SMulCommClass N α α] [IsScalarTower N α α]
variable [Module R α] [SMulCommClass R α α] [IsScalarTower R α α]
instance instSMul : SMul M (CentroidHom α) where
smul n f :=
{ (n • f : α →+ α) with
map_mul_left' := fun a b ↦ by
change n • f (a * b) = a * n • f b
rw [map_mul_left f, ← mul_smul_comm]
map_mul_right' := fun a b ↦ by
change n • f (a * b) = n • f a * b
rw [map_mul_right f, ← smul_mul_assoc] }
#noalign centroid_hom.has_nsmul
instance [SMul M N] [IsScalarTower M N α] : IsScalarTower M N (CentroidHom α) where
smul_assoc _ _ _ := ext <| fun _ => smul_assoc _ _ _
instance [SMulCommClass M N α] : SMulCommClass M N (CentroidHom α) where
smul_comm _ _ _ := ext <| fun _ => smul_comm _ _ _
instance [DistribMulAction Mᵐᵒᵖ α] [IsCentralScalar M α] : IsCentralScalar M (CentroidHom α) where
op_smul_eq_smul _ _ := ext <| fun _ => op_smul_eq_smul _ _
instance isScalarTowerRight : IsScalarTower M (CentroidHom α) (CentroidHom α) where
smul_assoc _ _ _ := rfl
instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_left' _ _)
map_mul_right' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_right' _ _) }⟩
#align centroid_hom.has_npow_nat CentroidHom.hasNPowNat
@[simp, norm_cast]
theorem coe_zero : ⇑(0 : CentroidHom α) = 0 :=
rfl
#align centroid_hom.coe_zero CentroidHom.coe_zero
@[simp, norm_cast]
theorem coe_one : ⇑(1 : CentroidHom α) = id :=
rfl
#align centroid_hom.coe_one CentroidHom.coe_one
@[simp, norm_cast]
theorem coe_add (f g : CentroidHom α) : ⇑(f + g) = f + g :=
rfl
#align centroid_hom.coe_add CentroidHom.coe_add
@[simp, norm_cast]
theorem coe_mul (f g : CentroidHom α) : ⇑(f * g) = f ∘ g :=
rfl
#align centroid_hom.coe_mul CentroidHom.coe_mul
@[simp, norm_cast]
theorem coe_smul (n : M) (f : CentroidHom α) : ⇑(n • f) = n • ⇑f :=
rfl
#align centroid_hom.coe_nsmul CentroidHom.coe_smul
@[simp]
theorem zero_apply (a : α) : (0 : CentroidHom α) a = 0 :=
rfl
#align centroid_hom.zero_apply CentroidHom.zero_apply
@[simp]
theorem one_apply (a : α) : (1 : CentroidHom α) a = a :=
rfl
#align centroid_hom.one_apply CentroidHom.one_apply
@[simp]
theorem add_apply (f g : CentroidHom α) (a : α) : (f + g) a = f a + g a :=
rfl
#align centroid_hom.add_apply CentroidHom.add_apply
@[simp]
theorem mul_apply (f g : CentroidHom α) (a : α) : (f * g) a = f (g a) :=
rfl
#align centroid_hom.mul_apply CentroidHom.mul_apply
@[simp]
theorem smul_apply (n : M) (f : CentroidHom α) (a : α) : (n • f) a = n • f a :=
rfl
#align centroid_hom.nsmul_apply CentroidHom.smul_apply
example : SMul ℕ (CentroidHom α) := instSMul
@[simp]
theorem toEnd_zero : (0 : CentroidHom α).toEnd = 0 :=
rfl
#align centroid_hom.to_End_zero CentroidHom.toEnd_zero
@[simp]
theorem toEnd_add (x y : CentroidHom α) : (x + y).toEnd = x.toEnd + y.toEnd :=
rfl
#align centroid_hom.to_End_add CentroidHom.toEnd_add
theorem toEnd_smul (m : M) (x : CentroidHom α) : (m • x).toEnd = m • x.toEnd :=
rfl
#align centroid_hom.to_End_nsmul CentroidHom.toEnd_smul
instance : AddCommMonoid (CentroidHom α) :=
coe_toAddMonoidHom_injective.addCommMonoid _ toEnd_zero toEnd_add (swap toEnd_smul)
instance : NatCast (CentroidHom α) where natCast n := n • (1 : CentroidHom α)
-- Porting note: `nolint simpNF` added because simplify fails on left-hand side
@[simp, norm_cast, nolint simpNF]
theorem coe_nat_cast (n : ℕ) : ⇑(n : CentroidHom α) = n • (CentroidHom.id α) :=
rfl
#align centroid_hom.coe_nat_cast CentroidHom.coe_nat_cast
theorem nat_cast_apply (n : ℕ) (m : α) : (n : CentroidHom α) m = n • m :=
rfl
#align centroid_hom.nat_cast_apply CentroidHom.nat_cast_apply
@[simp]
theorem toEnd_one : (1 : CentroidHom α).toEnd = 1 :=
rfl
#align centroid_hom.to_End_one CentroidHom.toEnd_one
@[simp]
theorem toEnd_mul (x y : CentroidHom α) : (x * y).toEnd = x.toEnd * y.toEnd :=
rfl
#align centroid_hom.to_End_mul CentroidHom.toEnd_mul
@[simp]
theorem toEnd_pow (x : CentroidHom α) (n : ℕ) : (x ^ n).toEnd = x.toEnd ^ n :=
rfl
#align centroid_hom.to_End_pow CentroidHom.toEnd_pow
@[simp, norm_cast]
theorem toEnd_nat_cast (n : ℕ) : (n : CentroidHom α).toEnd = ↑n :=
rfl
#align centroid_hom.to_End_nat_cast CentroidHom.toEnd_nat_cast
-- cf `add_monoid.End.semiring`
instance : Semiring (CentroidHom α) :=
toEnd_injective.semiring _ toEnd_zero toEnd_one toEnd_add toEnd_mul (swap toEnd_smul) toEnd_pow
toEnd_nat_cast
variable (α) in
/-- `CentroidHom.toEnd` as a `RingHom`. -/
@[simps]
def toEndRingHom : CentroidHom α →+* AddMonoid.End α where
toFun := toEnd
map_zero' := toEnd_zero
map_one' := toEnd_one
map_add' := toEnd_add
map_mul' := toEnd_mul
theorem comp_mul_comm (T S : CentroidHom α) (a b : α) : (T ∘ S) (a * b) = (S ∘ T) (a * b) := by
simp only [Function.comp_apply]
rw [map_mul_right, map_mul_left, ← map_mul_right, ← map_mul_left]
#align centroid_hom.comp_mul_comm CentroidHom.comp_mul_comm
instance : DistribMulAction M (CentroidHom α) :=
toEnd_injective.distribMulAction (toEndRingHom α).toAddMonoidHom toEnd_smul
instance : Module R (CentroidHom α) :=
toEnd_injective.module R (toEndRingHom α).toAddMonoidHom toEnd_smul
local notation "L" => AddMonoid.End.mulLeft
local notation "R" => AddMonoid.End.mulRight
lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
refine ⟨?_, fun h ↦ ?_⟩
· rintro ⟨f, rfl⟩ S (⟨a, rfl⟩ | ⟨b, rfl⟩)
· exact AddMonoidHom.ext fun b ↦ (map_mul_left f a b).symm
· exact AddMonoidHom.ext fun a ↦ (map_mul_right f a b).symm
· rw [Subsemiring.mem_centralizer_iff] at h
refine ⟨⟨T, fun a b ↦ ?_, fun a b ↦ ?_⟩, rfl⟩
· exact congr($(h (L a) (.inl ⟨a, rfl⟩)) b).symm
· exact congr($(h (R b) (.inr ⟨b, rfl⟩)) a).symm
/-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z :=
{ L (z : α) with
map_mul_left' := ((Set.mem_center_iff _).mp z.prop).left_comm
map_mul_right' := ((Set.mem_center_iff _).mp z.prop).left_assoc }
map_zero' := by
simp only [ZeroMemClass.coe_zero, map_zero]
exact rfl
map_add' := fun _ _ => by
simp only [AddSubmonoid.coe_add, NonUnitalSubsemiring.coe_toAddSubmonoid, map_add]
exact rfl
map_mul' := fun z₁ z₂ => by
ext a
exact (((Set.mem_center_iff _).mp z₁.prop).left_assoc z₂ a).symm
lemma centerToCentroid_apply (z : { x // x ∈ NonUnitalSubsemiring.center α }) (a : α) :
(centerToCentroid z) a = z * a := rfl
lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
have e2 (d : α) : T d = d * a := congr($(hT.trans hc) d)
constructor
case comm => exact (congr($hc ·))
case left_assoc => simpa [e1] using (map_mul_right T · ·)
case mid_assoc => exact fun b c ↦ by simpa [e1 c, e2 b] using
(map_mul_right T b c).symm.trans <| map_mul_left T b c
case right_assoc => simpa [e2] using (map_mul_left T · ·)
end NonUnitalNonAssocSemiring
section NonAssocSemiring
variable [NonAssocSemiring α]
/-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by refine ⟨?_, ?_, ?_, ?_⟩; all_goals simp [← map_mul_left, ← map_mul_right]⟩
left_inv := fun z ↦ Subtype.ext <| by simp [centerToCentroid_apply]
right_inv := fun T ↦ CentroidHom.ext <| by simp [centerToCentroid_apply, ← map_mul_right] }
end NonAssocSemiring
section NonUnitalNonAssocRing
variable [NonUnitalNonAssocRing α]
/-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) :=
⟨fun f ↦
{ (-f : α →+ α) with
map_mul_left' := fun a b ↦ by
change -f (a * b) = a * (-f b)
simp [map_mul_left]
map_mul_right' := fun a b ↦ by
change -f (a * b) = (-f a) * b
simp [map_mul_right] }⟩
instance : Sub (CentroidHom α) :=
⟨fun f g ↦
{ (f - g : α →+ α) with
map_mul_left' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = a * (⇑f - ⇑g) b
simp [map_mul_left, mul_sub]
map_mul_right' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = ((⇑f - ⇑g) a) * b
simp [map_mul_right, sub_mul] }⟩
#noalign centroid_hom.has_zsmul
instance : IntCast (CentroidHom α) where intCast z := z • (1 : CentroidHom α)
-- Porting note: `nolint simpNF` added because simplify fails on left-hand side
@[simp, norm_cast, nolint simpNF]
theorem coe_int_cast (z : ℤ) : ⇑(z : CentroidHom α) = z • (CentroidHom.id α) :=
rfl
#align centroid_hom.coe_int_cast CentroidHom.coe_int_cast
theorem int_cast_apply (z : ℤ) (m : α) : (z : CentroidHom α) m = z • m :=
rfl
#align centroid_hom.int_cast_apply CentroidHom.int_cast_apply
@[simp]
theorem toEnd_neg (x : CentroidHom α) : (-x).toEnd = -x.toEnd :=
rfl
#align centroid_hom.to_End_neg CentroidHom.toEnd_neg
@[simp]
theorem toEnd_sub (x y : CentroidHom α) : (x - y).toEnd = x.toEnd - y.toEnd :=
rfl
#align centroid_hom.to_End_sub CentroidHom.toEnd_sub
#align centroid_hom.to_End_zsmul CentroidHom.toEnd_smul
instance : AddCommGroup (CentroidHom α) :=
toEnd_injective.addCommGroup _
toEnd_zero toEnd_add toEnd_neg toEnd_sub (swap toEnd_smul) (swap toEnd_smul)
@[simp, norm_cast]
theorem coe_neg (f : CentroidHom α) : ⇑(-f) = -f :=
rfl
#align centroid_hom.coe_neg CentroidHom.coe_neg
@[simp, norm_cast]
theorem coe_sub (f g : CentroidHom α) : ⇑(f - g) = f - g :=
rfl
#align centroid_hom.coe_sub CentroidHom.coe_sub
@[simp]
theorem neg_apply (f : CentroidHom α) (a : α) : (-f) a = -f a :=
rfl
#align centroid_hom.neg_apply CentroidHom.neg_apply
@[simp]
theorem sub_apply (f g : CentroidHom α) (a : α) : (f - g) a = f a - g a :=
rfl
#align centroid_hom.sub_apply CentroidHom.sub_apply
@[simp, norm_cast]
theorem toEnd_int_cast (z : ℤ) : (z : CentroidHom α).toEnd = ↑z :=
rfl
#align centroid_hom.to_End_int_cast CentroidHom.toEnd_int_cast
instance instRing : Ring (CentroidHom α) :=
toEnd_injective.ring _ toEnd_zero toEnd_one toEnd_add toEnd_mul toEnd_neg toEnd_sub
(swap toEnd_smul) (swap toEnd_smul) toEnd_pow toEnd_nat_cast toEnd_int_cast
end NonUnitalNonAssocRing
section NonUnitalRing
variable [NonUnitalRing α]
-- Porting note: Not sure why Lean didn't like `CentroidHom.Ring`
-- See note [reducible non instances]
/-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) :=
{ CentroidHom.instRing with
mul_comm := fun f g ↦ by
ext
refine' sub_eq_zero.1 (or_self_iff.1 <| (h _ _) fun r ↦ _)
| rw [mul_assoc, sub_mul, sub_eq_zero, ← map_mul_right, ← map_mul_right, coe_mul, coe_mul,
comp_mul_comm] | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) :=
{ CentroidHom.instRing with
mul_comm := fun f g ↦ by
ext
refine' sub_eq_zero.1 (or_self_iff.1 <| (h _ _) fun r ↦ _)
| Mathlib.Algebra.Ring.CentroidHom.591_0.FQQ3LT1tg3cKlkH | /-- A prime associative ring has commutative centroid. -/
@[reducible]
def commRing (h : ∀ a b : α, (∀ r : α, a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
β : Type u_1
G : Type u_2
α : Type u_3
γ : Type u_4
inst✝³ : Group G
inst✝² : AddGroup α
inst✝¹ : SMul γ α
inst✝ : SlashAction β G α γ
k : β
g : G
a : α
⊢ (-a) ∣[k;γ] g + a ∣[k;γ] g = 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
| rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash] | @[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
| Mathlib.NumberTheory.ModularForms.SlashActions.58_0.3orIHeXinm1hkfX | @[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g | Mathlib_NumberTheory_ModularForms_SlashActions |
R : Type u_1
β : Type u_2
G : Type u_3
α : Type u_4
γ : Type u_5
inst✝⁷ : Group G
inst✝⁶ : AddGroup α
inst✝⁵ : Monoid γ
inst✝⁴ : MulAction γ α
inst✝³ : SMul R γ
inst✝² : SMul R α
inst✝¹ : IsScalarTower R γ α
inst✝ : SlashAction β G α γ
k : β
g : G
a : α
r : R
⊢ (r • a) ∣[k;γ] g = r • a ∣[k;γ] g | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
| rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul] | @[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
| Mathlib.NumberTheory.ModularForms.SlashActions.65_0.3orIHeXinm1hkfX | @[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g | Mathlib_NumberTheory_ModularForms_SlashActions |
β : Type u_1
G : Type u_2
H : Type u_3
α : Type u_4
γ : Type u_5
inst✝⁴ : Group G
inst✝³ : AddMonoid α
inst✝² : SMul γ α
inst✝¹ : Group H
inst✝ : SlashAction β G α γ
h : H →* G
k : β
a : α
⊢ (fun k g => SlashAction.map γ k (h g)) k 1 a = a | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by | simp only [map_one, SlashAction.slash_one] | /-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by | Mathlib.NumberTheory.ModularForms.SlashActions.75_0.3orIHeXinm1hkfX | /-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g | Mathlib_NumberTheory_ModularForms_SlashActions |
β : Type u_1
G : Type u_2
H : Type u_3
α : Type u_4
γ : Type u_5
inst✝⁴ : Group G
inst✝³ : AddMonoid α
inst✝² : SMul γ α
inst✝¹ : Group H
inst✝ : SlashAction β G α γ
h : H →* G
k : β
g gg : H
a : α
⊢ (fun k g => SlashAction.map γ k (h g)) k (g * gg) a =
(fun k g => SlashAction.map γ k (h g)) k gg ((fun k g => SlashAction.map γ k (h g)) k g a) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by | simp only [map_mul, SlashAction.slash_mul] | /-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by | Mathlib.NumberTheory.ModularForms.SlashActions.75_0.3orIHeXinm1hkfX | /-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
⊢ f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
| ext1 x | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
⊢ (f ∣[k](A * B)) x = ((f ∣[k]A) ∣[k]B) x | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
| simp_rw [slash, UpperHalfPlane.denom_cocycle A B x] | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
⊢ f ((A * B) • x) * ↑(Matrix.det ↑↑(A * B)) ^ (k - 1) * (denom A (smulAux B x) * denom B x) ^ (-k) =
f (A • B • x) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A (B • x) ^ (-k) * ↑(Matrix.det ↑↑B) ^ (k - 1) *
denom B x ^ (-k) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
| have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
⊢ (A * B) • x = A • B • x | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by | convert UpperHalfPlane.mul_smul' A B x | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by | Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
⊢ f ((A * B) • x) * ↑(Matrix.det ↑↑(A * B)) ^ (k - 1) * (denom A (smulAux B x) * denom B x) ^ (-k) =
f (A • B • x) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A (B • x) ^ (-k) * ↑(Matrix.det ↑↑B) ^ (k - 1) *
denom B x ^ (-k) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
| rw [e3] | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
⊢ f (A • B • x) * ↑(Matrix.det ↑↑(A * B)) ^ (k - 1) * (denom A (smulAux B x) * denom B x) ^ (-k) =
f (A • B • x) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A (B • x) ^ (-k) * ↑(Matrix.det ↑↑B) ^ (k - 1) *
denom B x ^ (-k) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
| simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at * | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
⊢ f (A • B • x) * (↑(Matrix.det ↑↑A) * ↑(Matrix.det ↑↑B)) ^ (k - 1) *
((↑(↑↑A 1 0) * ((↑(↑↑B 0 0) * ↑x + ↑(↑↑B 0 1)) / (↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1))) + ↑(↑↑A 1 1)) *
(↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1))) ^
(-k) =
f (A • B • x) * ↑(Matrix.det ↑↑A) ^ (k - 1) *
(↑(↑↑A 1 0) * ((↑(↑↑B 0 0) * ↑x + ↑(↑↑B 0 1)) / (↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1))) + ↑(↑↑A 1 1)) ^ (-k) *
↑(Matrix.det ↑↑B) ^ (k - 1) *
(↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1)) ^ (-k) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
| field_simp | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
⊢ f (A • B • x) * (↑(Matrix.det ↑↑A) * ↑(Matrix.det ↑↑B)) ^ (k - 1) /
((↑(↑↑A 1 0) * (↑(↑↑B 0 0) * ↑x + ↑(↑↑B 0 1)) / (↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1)) + ↑(↑↑A 1 1)) *
(↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1))) ^
k =
f (A • B • x) * ↑(Matrix.det ↑↑A) ^ (k - 1) * ↑(Matrix.det ↑↑B) ^ (k - 1) /
((↑(↑↑A 1 0) * (↑(↑↑B 0 0) * ↑x + ↑(↑↑B 0 1)) / (↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1)) + ↑(↑↑A 1 1)) ^ k *
(↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1)) ^ k) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
| have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow] | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
⊢ (↑(Matrix.det ↑↑A) * ↑(Matrix.det ↑↑B)) ^ (k - 1) = ↑(Matrix.det ↑↑A) ^ (k - 1) * ↑(Matrix.det ↑↑B) ^ (k - 1) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
| simp_rw [← mul_zpow] | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A B : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
x : ℍ
e3 : (A * B) • x = A • B • x
this : (↑(Matrix.det ↑↑A) * ↑(Matrix.det ↑↑B)) ^ (k - 1) = ↑(Matrix.det ↑↑A) ^ (k - 1) * ↑(Matrix.det ↑↑B) ^ (k - 1)
⊢ f (A • B • x) * (↑(Matrix.det ↑↑A) * ↑(Matrix.det ↑↑B)) ^ (k - 1) /
((↑(↑↑A 1 0) * (↑(↑↑B 0 0) * ↑x + ↑(↑↑B 0 1)) / (↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1)) + ↑(↑↑A 1 1)) *
(↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1))) ^
k =
f (A • B • x) * ↑(Matrix.det ↑↑A) ^ (k - 1) * ↑(Matrix.det ↑↑B) ^ (k - 1) /
((↑(↑↑A 1 0) * (↑(↑↑B 0 0) * ↑x + ↑(↑↑B 0 1)) / (↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1)) + ↑(↑↑A 1 1)) ^ k *
(↑(↑↑B 1 0) * ↑x + ↑(↑↑B 1 1)) ^ k) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
| simp_rw [this, ← mul_assoc, ← mul_zpow] | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
| Mathlib.NumberTheory.ModularForms.SlashActions.102_0.3orIHeXinm1hkfX | private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
⊢ (f + g) ∣[k]A = f ∣[k]A + g ∣[k]A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
| ext1 | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
| Mathlib.NumberTheory.ModularForms.SlashActions.119_0.3orIHeXinm1hkfX | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x✝ : ℍ
⊢ ((f + g) ∣[k]A) x✝ = (f ∣[k]A + g ∣[k]A) x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
| simp only [slash, Pi.add_apply, denom, zpow_neg] | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
| Mathlib.NumberTheory.ModularForms.SlashActions.119_0.3orIHeXinm1hkfX | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x✝ : ℍ
⊢ (f (A • x✝) + g (A • x✝)) * ↑(Matrix.det ↑↑A) ^ (k - 1) * ((↑(↑↑A 1 0) * ↑x✝ + ↑(↑↑A 1 1)) ^ k)⁻¹ =
f (A • x✝) * ↑(Matrix.det ↑↑A) ^ (k - 1) * ((↑(↑↑A 1 0) * ↑x✝ + ↑(↑↑A 1 1)) ^ k)⁻¹ +
g (A • x✝) * ↑(Matrix.det ↑↑A) ^ (k - 1) * ((↑(↑↑A 1 0) * ↑x✝ + ↑(↑↑A 1 1)) ^ k)⁻¹ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
| ring | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
| Mathlib.NumberTheory.ModularForms.SlashActions.119_0.3orIHeXinm1hkfX | private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
f : ℍ → ℂ
⊢ ∀ (x : ℍ), (f ∣[k]1) x = f x | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by | simp [slash, denom] | private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by | Mathlib.NumberTheory.ModularForms.SlashActions.125_0.3orIHeXinm1hkfX | private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
c : α
⊢ (c • f) ∣[k]A = c • f ∣[k]A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
| simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)] | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
| Mathlib.NumberTheory.ModularForms.SlashActions.130_0.3orIHeXinm1hkfX | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
c : α
⊢ ((c • 1) • f) ∣[k]A = (c • 1) • f ∣[k]A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
| ext1 | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
| Mathlib.NumberTheory.ModularForms.SlashActions.130_0.3orIHeXinm1hkfX | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
c : α
x✝ : ℍ
⊢ (((c • 1) • f) ∣[k]A) x✝ = ((c • 1) • f ∣[k]A) x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
| simp_rw [slash] | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
| Mathlib.NumberTheory.ModularForms.SlashActions.130_0.3orIHeXinm1hkfX | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
c : α
x✝ : ℍ
⊢ ((c • 1) • f) (A • x✝) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A x✝ ^ (-k) = ((c • 1) • f ∣[k]A) x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
| simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply] | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
| Mathlib.NumberTheory.ModularForms.SlashActions.130_0.3orIHeXinm1hkfX | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f : ℍ → ℂ
c : α
x✝ : ℍ
⊢ c • 1 * f (A • x✝) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A x✝ ^ (-k) =
c • 1 * (f (A • x✝) * ↑(Matrix.det ↑↑A) ^ (k - 1) * denom A x✝ ^ (-k)) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
| ring | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
| Mathlib.NumberTheory.ModularForms.SlashActions.130_0.3orIHeXinm1hkfX | private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f : ℍ → ℂ
α : Type u_1
inst✝¹ : SMul α ℂ
inst✝ : IsScalarTower α ℂ ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
x✝ : ℍ
⊢ (0 ∣[k]A) x✝ = OfNat.ofNat 0 x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by | simp only [slash, Pi.zero_apply, zero_mul] | private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by | Mathlib.NumberTheory.ModularForms.SlashActions.138_0.3orIHeXinm1hkfX | private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f : ℍ → ℂ
A : SL(2, ℤ)
⊢ 1 ∣[0] A = 1 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
| have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe' | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
| Mathlib.NumberTheory.ModularForms.SlashActions.179_0.3orIHeXinm1hkfX | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f : ℍ → ℂ
A : SL(2, ℤ)
this : Matrix.det ↑↑↑A = 1
⊢ 1 ∣[0] A = 1 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
| funext | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
| Mathlib.NumberTheory.ModularForms.SlashActions.179_0.3orIHeXinm1hkfX | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f : ℍ → ℂ
A : SL(2, ℤ)
this : Matrix.det ↑↑↑A = 1
x✝ : ℍ
⊢ (1 ∣[0] A) x✝ = OfNat.ofNat 1 x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
| rw [SL_slash, slash_def, slash, zero_sub, this] | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
| Mathlib.NumberTheory.ModularForms.SlashActions.179_0.3orIHeXinm1hkfX | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f : ℍ → ℂ
A : SL(2, ℤ)
this : Matrix.det ↑↑↑A = 1
x✝ : ℍ
⊢ OfNat.ofNat 1 (↑A • x✝) * ↑1 ^ (-1) * denom (↑A) x✝ ^ (-0) = OfNat.ofNat 1 x✝ | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
| simp | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
| Mathlib.NumberTheory.ModularForms.SlashActions.179_0.3orIHeXinm1hkfX | /-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ (f ∣[k] γ) z = f z ↔ f (γ • z) = (↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k * f z | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
| simp only [subgroup_slash, slash_def, ModularForm.slash] | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
| Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ f (↑↑γ • z) * ↑(Matrix.det ↑↑↑↑γ) ^ (k - 1) * denom (↑↑γ) z ^ (-k) = f z ↔
f (γ • z) = (↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k * f z | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
| convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2 | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
| Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z | Mathlib_NumberTheory_ModularForms_SlashActions |
case h.e'_1.h.e'_2
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ f (↑↑γ • z) * ↑(Matrix.det ↑↑↑↑γ) ^ (k - 1) * denom (↑↑γ) z ^ (-k) =
((↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k)⁻¹ * f (γ • z) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· | rw [mul_comm] | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· | Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z | Mathlib_NumberTheory_ModularForms_SlashActions |
case h.e'_1.h.e'_2
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ denom (↑↑γ) z ^ (-k) * (f (↑↑γ • z) * ↑(Matrix.det ↑↑↑↑γ) ^ (k - 1)) =
((↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k)⁻¹ * f (γ • z) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
| simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb] | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
| Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z | Mathlib_NumberTheory_ModularForms_SlashActions |
case h.e'_1.h.e'_2
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ ((↑(↑↑↑↑γ 1 0) * ↑z + ↑(↑↑↑↑γ 1 1)) ^ k)⁻¹ * f (↑↑γ • z) = ((↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k)⁻¹ * f (↑↑γ • z) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
| rfl | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
| Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z | Mathlib_NumberTheory_ModularForms_SlashActions |
case convert_4
Γ✝ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
Γ : Subgroup SL(2, ℤ)
f : ℍ → ℂ
γ : ↥Γ
z : ℍ
⊢ (↑(↑↑↑γ 1 0) * ↑z + ↑(↑↑↑γ 1 1)) ^ k ≠ 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· | convert zpow_ne_zero k (denom_ne_zero γ z) | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· | Mathlib.NumberTheory.ModularForms.SlashActions.188_0.3orIHeXinm1hkfX | /-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
⊢ (f * g) ∣[k1 + k2] A = Matrix.det ↑↑A • f ∣[k1] A * g ∣[k2] A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
| ext1 x | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
⊢ ((f * g) ∣[k1 + k2] A) x = (Matrix.det ↑↑A • f ∣[k1] A * g ∣[k2] A) x | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
| simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul] | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
⊢ f (A • x) * g (A • x) * ↑(Matrix.det ↑↑A) ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) =
↑(Matrix.det ↑↑A) * (f (A • x) * ↑(Matrix.det ↑↑A) ^ (k1 - 1) * denom A x ^ (-k1)) *
(g (A • x) * ↑(Matrix.det ↑↑A) ^ (k2 - 1) * denom A x ^ (-k2)) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
| set d : ℂ := ↑((↑ₘA).det : ℝ) | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
⊢ f (A • x) * g (A • x) * d ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) =
d * (f (A • x) * d ^ (k1 - 1) * denom A x ^ (-k1)) * (g (A • x) * d ^ (k2 - 1) * denom A x ^ (-k2)) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
| have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
⊢ d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
| have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
⊢ d ≠ 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
| dsimp | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
⊢ ¬↑(Matrix.det ↑↑A) = 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
| norm_cast | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
⊢ ¬Matrix.det ↑↑A = 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
| exact Matrix.GLPos.det_ne_zero A | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
this : d ≠ 0
⊢ d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
| rw [← zpow_one_add₀ this, ← zpow_add₀ this] | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
this : d ≠ 0
⊢ d ^ (k1 + k2 - 1) = d ^ (1 + (k1 - 1) + (k2 - 1)) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
| congr | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case e_a
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
this : d ≠ 0
⊢ k1 + k2 - 1 = 1 + (k1 - 1) + (k2 - 1) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; | ring | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; | Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)
⊢ f (A • x) * g (A • x) * d ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) =
d * (f (A • x) * d ^ (k1 - 1) * denom A x ^ (-k1)) * (g (A • x) * d ^ (k2 - 1) * denom A x ^ (-k2)) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
| have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
rw [Int.neg_add, zpow_add₀]
exact UpperHalfPlane.denom_ne_zero A x | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)
⊢ denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
| rw [Int.neg_add, zpow_add₀] | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case ha
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)
⊢ denom A x ≠ 0 | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
rw [Int.neg_add, zpow_add₀]
| exact UpperHalfPlane.denom_ne_zero A x | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
rw [Int.neg_add, zpow_add₀]
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)
h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2)
⊢ f (A • x) * g (A • x) * d ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) =
d * (f (A • x) * d ^ (k1 - 1) * denom A x ^ (-k1)) * (g (A • x) * d ^ (k2 - 1) * denom A x ^ (-k2)) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
rw [Int.neg_add, zpow_add₀]
exact UpperHalfPlane.denom_ne_zero A x
| rw [h1, h22] | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
rw [Int.neg_add, zpow_add₀]
exact UpperHalfPlane.denom_ne_zero A x
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
case h
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(Matrix.det ↑↑A)
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)
h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2)
⊢ f (A • x) * g (A • x) * (d * d ^ (k1 - 1) * d ^ (k2 - 1)) * (denom A x ^ (-k1) * denom A x ^ (-k2)) =
d * (f (A • x) * d ^ (k1 - 1) * denom A x ^ (-k1)) * (g (A • x) * d ^ (k2 - 1) * denom A x ^ (-k2)) | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
rw [Int.neg_add, zpow_add₀]
exact UpperHalfPlane.denom_ne_zero A x
rw [h1, h22]
| ring | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
rw [Int.neg_add, zpow_add₀]
exact UpperHalfPlane.denom_ne_zero A x
rw [h1, h22]
| Mathlib.NumberTheory.ModularForms.SlashActions.202_0.3orIHeXinm1hkfX | theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : SL(2, ℤ)
f g : ℍ → ℂ
⊢ (f * g) ∣[k1 + k2] ↑A = Matrix.det ↑↑↑A • f ∣[k1] A * g ∣[k2] A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
rw [Int.neg_add, zpow_add₀]
exact UpperHalfPlane.denom_ne_zero A x
rw [h1, h22]
ring
#align modular_form.mul_slash ModularForm.mul_slash
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A :=
calc
(f * g) ∣[k1 + k2] (A : GL(2, ℝ)⁺) =
((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
| apply mul_slash | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A :=
calc
(f * g) ∣[k1 + k2] (A : GL(2, ℝ)⁺) =
((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
| Mathlib.NumberTheory.ModularForms.SlashActions.223_0.3orIHeXinm1hkfX | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : SL(2, ℤ)
f g : ℍ → ℂ
⊢ Matrix.det ↑↑↑A • f ∣[k1] A * g ∣[k2] A = 1 • f ∣[k1] A * g ∣[k2] A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
rw [Int.neg_add, zpow_add₀]
exact UpperHalfPlane.denom_ne_zero A x
rw [h1, h22]
ring
#align modular_form.mul_slash ModularForm.mul_slash
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A :=
calc
(f * g) ∣[k1 + k2] (A : GL(2, ℝ)⁺) =
((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
apply mul_slash
_ = (1 : ℝ) • f ∣[k1] A * g ∣[k2] A := by | rw [det_coe'] | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A :=
calc
(f * g) ∣[k1 + k2] (A : GL(2, ℝ)⁺) =
((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
apply mul_slash
_ = (1 : ℝ) • f ∣[k1] A * g ∣[k2] A := by | Mathlib.NumberTheory.ModularForms.SlashActions.223_0.3orIHeXinm1hkfX | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : SL(2, ℤ)
f g : ℍ → ℂ
⊢ 1 • f ∣[k1] A * g ∣[k2] A = f ∣[k1] A * g ∣[k2] A | /-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45ce799d18fd44248"
/-!
# Slash actions
This file defines a class of slash actions, which are families of right actions of a given group
parametrized by some Type. This is modeled on the slash action of `GLPos (Fin 2) ℝ` on the space
of modular forms.
## Notation
In the `ModularForm` locale, this provides
* `f ∣[k;γ] A`: the `k`th `γ`-compatible slash action by `A` on `f`
* `f ∣[k] A`: the `k`th `ℂ`-compatible slash action by `A` on `f`; a shorthand for `f ∣[k;ℂ] A`
-/
open Complex UpperHalfPlane
open scoped UpperHalfPlane
local notation "GL(" n ", " R ")" "⁺" => Matrix.GLPos (Fin n) R
local notation "SL(" n ", " R ")" => Matrix.SpecialLinearGroup (Fin n) R
local notation:1024 "↑ₘ" A:1024 =>
(((A : GL(2, ℝ)⁺) : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) _)
-- like `↑ₘ`, but allows the user to specify the ring `R`. Useful to help Lean elaborate.
local notation:1024 "↑ₘ[" R "]" A:1024 =>
((A : GL (Fin 2) R) : Matrix (Fin 2) (Fin 2) R)
/-- A general version of the slash action of the space of modular forms.-/
class SlashAction (β G α γ : Type*) [Group G] [AddMonoid α] [SMul γ α] where
map : β → G → α → α
zero_slash : ∀ (k : β) (g : G), map k g 0 = 0
slash_one : ∀ (k : β) (a : α), map k 1 a = a
slash_mul : ∀ (k : β) (g h : G) (a : α), map k (g * h) a = map k h (map k g a)
smul_slash : ∀ (k : β) (g : G) (a : α) (z : γ), map k g (z • a) = z • map k g a
add_slash : ∀ (k : β) (g : G) (a b : α), map k g (a + b) = map k g a + map k g b
#align slash_action SlashAction
scoped[ModularForm] notation:100 f " ∣[" k ";" γ "] " a:100 => SlashAction.map γ k a f
scoped[ModularForm] notation:100 f " ∣[" k "] " a:100 => SlashAction.map ℂ k a f
open scoped ModularForm
@[simp]
theorem SlashAction.neg_slash {β G α γ : Type*} [Group G] [AddGroup α] [SMul γ α]
[SlashAction β G α γ] (k : β) (g : G) (a : α) : (-a) ∣[k;γ] g = -a ∣[k;γ] g :=
eq_neg_of_add_eq_zero_left <| by
rw [← SlashAction.add_slash, add_left_neg, SlashAction.zero_slash]
#align slash_action.neg_slash SlashAction.neg_slash
@[simp]
theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
#align slash_action.smul_slash_of_tower SlashAction.smul_slash_of_tower
attribute [simp] SlashAction.zero_slash SlashAction.slash_one SlashAction.smul_slash
SlashAction.add_slash
/-- Slash_action induced by a monoid homomorphism.-/
def monoidHomSlashAction {β G H α γ : Type*} [Group G] [AddMonoid α] [SMul γ α] [Group H]
[SlashAction β G α γ] (h : H →* G) : SlashAction β H α γ where
map k g := SlashAction.map γ k (h g)
zero_slash k g := SlashAction.zero_slash k (h g)
slash_one k a := by simp only [map_one, SlashAction.slash_one]
slash_mul k g gg a := by simp only [map_mul, SlashAction.slash_mul]
smul_slash _ _ := SlashAction.smul_slash _ _
add_slash _ g _ _ := SlashAction.add_slash _ (h g) _ _
#align monoid_hom_slash_action monoidHomSlashAction
namespace ModularForm
noncomputable section
/-- The weight `k` action of `GL(2, ℝ)⁺` on functions `f : ℍ → ℂ`. -/
def slash (k : ℤ) (γ : GL(2, ℝ)⁺) (f : ℍ → ℂ) (x : ℍ) : ℂ :=
f (γ • x) * (((↑ₘγ).det : ℝ) : ℂ) ^ (k - 1) * UpperHalfPlane.denom γ x ^ (-k)
#align modular_form.slash ModularForm.slash
variable {Γ : Subgroup SL(2, ℤ)} {k : ℤ} (f : ℍ → ℂ)
section
-- temporary notation until the instance is built
local notation:100 f " ∣[" k "]" γ:100 => ModularForm.slash k γ f
private theorem slash_mul (k : ℤ) (A B : GL(2, ℝ)⁺) (f : ℍ → ℂ) :
f ∣[k](A * B) = (f ∣[k]A) ∣[k]B := by
ext1 x
simp_rw [slash, UpperHalfPlane.denom_cocycle A B x]
have e3 : (A * B) • x = A • B • x := by convert UpperHalfPlane.mul_smul' A B x
rw [e3]
simp only [UpperHalfPlane.num, UpperHalfPlane.denom, ofReal_mul, Subgroup.coe_mul,
UpperHalfPlane.coe_smul, Units.val_mul, Matrix.det_mul,
UpperHalfPlane.smulAux, UpperHalfPlane.smulAux', UpperHalfPlane.coe_mk] at *
field_simp
have : (((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ)) ^ (k - 1) =
((↑(↑A : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) *
((↑(↑B : GL (Fin 2) ℝ) : Matrix (Fin 2) (Fin 2) ℝ).det : ℂ) ^ (k - 1) := by
simp_rw [← mul_zpow]
simp_rw [this, ← mul_assoc, ← mul_zpow]
private theorem add_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f + g) ∣[k]A = f ∣[k]A + g ∣[k]A := by
ext1
simp only [slash, Pi.add_apply, denom, zpow_neg]
ring
private theorem slash_one (k : ℤ) (f : ℍ → ℂ) : f ∣[k]1 = f :=
funext <| by simp [slash, denom]
variable {α : Type*} [SMul α ℂ] [IsScalarTower α ℂ ℂ]
private theorem smul_slash (k : ℤ) (A : GL(2, ℝ)⁺) (f : ℍ → ℂ) (c : α) :
(c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
funext fun _ => by simp only [slash, Pi.zero_apply, zero_mul]
instance : SlashAction ℤ GL(2, ℝ)⁺ (ℍ → ℂ) ℂ where
map := slash
zero_slash := zero_slash
slash_one := slash_one
slash_mul := slash_mul
smul_slash := smul_slash
add_slash := add_slash
end
theorem slash_def (A : GL(2, ℝ)⁺) : f ∣[k] A = slash k A f :=
rfl
#align modular_form.slash_def ModularForm.slash_def
instance subgroupAction (Γ : Subgroup SL(2, ℤ)) : SlashAction ℤ Γ (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(MonoidHom.comp (Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) (Subgroup.subtype Γ)))
#align modular_form.subgroup_action ModularForm.subgroupAction
@[simp]
theorem subgroup_slash (Γ : Subgroup SL(2, ℤ)) (γ : Γ) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
#align modular_form.subgroup_slash ModularForm.subgroup_slash
instance SLAction : SlashAction ℤ SL(2, ℤ) (ℍ → ℂ) ℂ :=
monoidHomSlashAction
(MonoidHom.comp Matrix.SpecialLinearGroup.toGLPos
(Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)))
set_option linter.uppercaseLean3 false in
#align modular_form.SL_action ModularForm.SLAction
@[simp]
theorem SL_slash (γ : SL(2, ℤ)) : f ∣[k] γ = f ∣[k] (γ : GL(2, ℝ)⁺) :=
rfl
set_option linter.uppercaseLean3 false in
#align modular_form.SL_slash ModularForm.SL_slash
/-- The constant function 1 is invariant under any element of `SL(2, ℤ)`. -/
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem is_invariant_one (A : SL(2, ℤ)) : (1 : ℍ → ℂ) ∣[(0 : ℤ)] A = (1 : ℍ → ℂ) := by
have : ((↑ₘ(A : GL(2, ℝ)⁺)).det : ℝ) = 1 := det_coe'
funext
rw [SL_slash, slash_def, slash, zero_sub, this]
simp
#align modular_form.is_invariant_one ModularForm.is_invariant_one
/-- A function `f : ℍ → ℂ` is slash-invariant, of weight `k ∈ ℤ` and level `Γ`,
if for every matrix `γ ∈ Γ` we have `f(γ • z)= (c*z+d)^k f(z)` where `γ= ![![a, b], ![c, d]]`,
and it acts on `ℍ` via Möbius transformations. -/
theorem slash_action_eq'_iff (k : ℤ) (Γ : Subgroup SL(2, ℤ)) (f : ℍ → ℂ) (γ : Γ) (z : ℍ) :
(f ∣[k] γ) z = f z ↔ f (γ • z) = ((↑ₘ[ℤ] γ 1 0 : ℂ) * z + (↑ₘ[ℤ] γ 1 1 : ℂ)) ^ k * f z := by
simp only [subgroup_slash, slash_def, ModularForm.slash]
convert inv_mul_eq_iff_eq_mul₀ (G₀ := ℂ) _ using 2
· rw [mul_comm]
simp only [denom, zpow_neg, det_coe', ofReal_one, one_zpow, mul_one, subgroup_to_sl_moeb,
sl_moeb]
rfl
· convert zpow_ne_zero k (denom_ne_zero γ z)
#align modular_form.slash_action_eq'_iff ModularForm.slash_action_eq'_iff
theorem mul_slash (k1 k2 : ℤ) (A : GL(2, ℝ)⁺) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = ((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
ext1 x
simp only [slash_def, slash, Matrix.GeneralLinearGroup.val_det_apply,
Pi.mul_apply, Pi.smul_apply, Algebra.smul_mul_assoc, real_smul]
set d : ℂ := ↑((↑ₘA).det : ℝ)
have h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1) := by
have : d ≠ 0 := by
dsimp
norm_cast
exact Matrix.GLPos.det_ne_zero A
rw [← zpow_one_add₀ this, ← zpow_add₀ this]
congr; ring
have h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2) := by
rw [Int.neg_add, zpow_add₀]
exact UpperHalfPlane.denom_ne_zero A x
rw [h1, h22]
ring
#align modular_form.mul_slash ModularForm.mul_slash
-- @[simp] -- Porting note: simpNF says LHS simplifies to something more complex
theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A :=
calc
(f * g) ∣[k1 + k2] (A : GL(2, ℝ)⁺) =
((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
apply mul_slash
_ = (1 : ℝ) • f ∣[k1] A * g ∣[k2] A := by rw [det_coe']
_ = f ∣[k1] A * g ∣[k2] A := by | rw [one_smul] | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A :=
calc
(f * g) ∣[k1 + k2] (A : GL(2, ℝ)⁺) =
((↑ₘA).det : ℝ) • f ∣[k1] A * g ∣[k2] A := by
apply mul_slash
_ = (1 : ℝ) • f ∣[k1] A * g ∣[k2] A := by rw [det_coe']
_ = f ∣[k1] A * g ∣[k2] A := by | Mathlib.NumberTheory.ModularForms.SlashActions.223_0.3orIHeXinm1hkfX | theorem mul_slash_SL2 (k1 k2 : ℤ) (A : SL(2, ℤ)) (f g : ℍ → ℂ) :
(f * g) ∣[k1 + k2] A = f ∣[k1] A * g ∣[k2] A | Mathlib_NumberTheory_ModularForms_SlashActions |
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P✝ : Type u_3
inst✝ : CommSemiring P✝
P : Ideal R
hp : IsPrime P
⊢ 1 ∈ { carrier := (↑P)ᶜ, mul_mem' := (_ : ∀ {x y : R}, x ∈ (↑P)ᶜ → y ∈ (↑P)ᶜ → x * y ∈ ↑P → False) }.carrier | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by | convert P.ne_top_iff_one.1 hp.1 | /-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by | Mathlib.RingTheory.Localization.AtPrime.45_0.QSwWrbtcZl7L7lq | /-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
hze : 0 = 1
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
| rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
| Mathlib.RingTheory.Localization.AtPrime.72_0.QSwWrbtcZl7L7lq | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
hze : (algebraMap R S) 0 = (algebraMap R S) 1
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
| obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
| Mathlib.RingTheory.Localization.AtPrime.72_0.QSwWrbtcZl7L7lq | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S | Mathlib_RingTheory_Localization_AtPrime |
case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
hze : (algebraMap R S) 0 = (algebraMap R S) 1
t : ↥(Ideal.primeCompl P)
ht : ↑t * 0 = ↑t * 1
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
| have htz : (t : R) = 0 := by simpa using ht.symm | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
| Mathlib.RingTheory.Localization.AtPrime.72_0.QSwWrbtcZl7L7lq | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
hze : (algebraMap R S) 0 = (algebraMap R S) 1
t : ↥(Ideal.primeCompl P)
ht : ↑t * 0 = ↑t * 1
⊢ ↑t = 0 | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by | simpa using ht.symm | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by | Mathlib.RingTheory.Localization.AtPrime.72_0.QSwWrbtcZl7L7lq | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S | Mathlib_RingTheory_Localization_AtPrime |
case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
hze : (algebraMap R S) 0 = (algebraMap R S) 1
t : ↥(Ideal.primeCompl P)
ht : ↑t * 0 = ↑t * 1
htz : ↑t = 0
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
| exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P) | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
| Mathlib.RingTheory.Localization.AtPrime.72_0.QSwWrbtcZl7L7lq | theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this : _root_.Nontrivial S := Nontrivial S P
⊢ ∀ (a b : S), a ∈ nonunits S → b ∈ nonunits S → a + b ∈ nonunits S | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
| intro x y hx hy hu | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonunits S
hy : y ∈ nonunits S
hu : IsUnit (x + y)
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
| cases' isUnit_iff_exists_inv.1 hu with z hxyz | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonunits S
hy : y ∈ nonunits S
hu : IsUnit (x + y)
z : S
hxyz : (x + y) * z = 1
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
| have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩ | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonunits S
hy : y ∈ nonunits S
hu : IsUnit (x + y)
z : S
hxyz : (x + y) * z = 1
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
| rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩ | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonunits S
hy : y ∈ nonunits S
hu : IsUnit (x + y)
z : S
hxyz : (x + y) * z = 1
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hrx : mk' S rx sx = x
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
| rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩ | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonunits S
hy : y ∈ nonunits S
hu : IsUnit (x + y)
z : S
hxyz : (x + y) * z = 1
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hrx : mk' S rx sx = x
ry : R
sy : ↥(Ideal.primeCompl P)
hry : mk' S ry sy = y
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
| rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩ | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonunits S
hy : y ∈ nonunits S
hu : IsUnit (x + y)
z : S
hxyz : (x + y) * z = 1
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hrx : mk' S rx sx = x
ry : R
sy : ↥(Ideal.primeCompl P)
hry : mk' S ry sy = y
rz : R
sz : ↥(Ideal.primeCompl P)
hrz : mk' S rz sz = z
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
| rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hx : x ∈ nonunits S
hy : y ∈ nonunits S
hu : IsUnit (x + y)
z : S
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hrx : mk' S rx sx = x
ry : R
sy : ↥(Ideal.primeCompl P)
hry : mk' S ry sy = y
rz : R
sz : ↥(Ideal.primeCompl P)
hxyz :
mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 { val := 1, property := (_ : 1 ∈ Ideal.primeCompl P) }
hrz : mk' S rz sz = z
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
| rw [← hrx] at hx | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hy : y ∈ nonunits S
hu : IsUnit (x + y)
z : S
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hx : mk' S rx sx ∈ nonunits S
hrx : mk' S rx sx = x
ry : R
sy : ↥(Ideal.primeCompl P)
hry : mk' S ry sy = y
rz : R
sz : ↥(Ideal.primeCompl P)
hxyz :
mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 { val := 1, property := (_ : 1 ∈ Ideal.primeCompl P) }
hrz : mk' S rz sz = z
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
| rw [← hry] at hy | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hu : IsUnit (x + y)
z : S
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hx : mk' S rx sx ∈ nonunits S
hrx : mk' S rx sx = x
ry : R
sy : ↥(Ideal.primeCompl P)
hy : mk' S ry sy ∈ nonunits S
hry : mk' S ry sy = y
rz : R
sz : ↥(Ideal.primeCompl P)
hxyz :
mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 { val := 1, property := (_ : 1 ∈ Ideal.primeCompl P) }
hrz : mk' S rz sz = z
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
| obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hu : IsUnit (x + y)
z : S
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hx : mk' S rx sx ∈ nonunits S
hrx : mk' S rx sx = x
ry : R
sy : ↥(Ideal.primeCompl P)
hy : mk' S ry sy ∈ nonunits S
hry : mk' S ry sy = y
rz : R
sz : ↥(Ideal.primeCompl P)
hxyz :
mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 { val := 1, property := (_ : 1 ∈ Ideal.primeCompl P) }
hrz : mk' S rz sz = z
t : ↥(Ideal.primeCompl P)
ht :
↑t * (↑{ val := 1, property := (_ : 1 ∈ Ideal.primeCompl P) } * ((rx * ↑sy + ry * ↑sx) * rz)) =
↑t * (↑(sx * sy * sz) * 1)
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
| simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hu : IsUnit (x + y)
z : S
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hx : mk' S rx sx ∈ nonunits S
hrx : mk' S rx sx = x
ry : R
sy : ↥(Ideal.primeCompl P)
hy : mk' S ry sy ∈ nonunits S
hry : mk' S ry sy = y
rz : R
sz : ↥(Ideal.primeCompl P)
hxyz :
mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 { val := 1, property := (_ : 1 ∈ Ideal.primeCompl P) }
hrz : mk' S rz sz = z
t : ↥(Ideal.primeCompl P)
ht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)
⊢ False | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
| suffices : (t : R) * (sx * sy * sz) ∈ P | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝¹ : _root_.Nontrivial S := Nontrivial S P
x y : S
hu : IsUnit (x + y)
z : S
this✝ : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hx : mk' S rx sx ∈ nonunits S
hrx : mk' S rx sx = x
ry : R
sy : ↥(Ideal.primeCompl P)
hy : mk' S ry sy ∈ nonunits S
hry : mk' S ry sy = y
rz : R
sz : ↥(Ideal.primeCompl P)
hxyz :
mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 { val := 1, property := (_ : 1 ∈ Ideal.primeCompl P) }
hrz : mk' S rz sz = z
t : ↥(Ideal.primeCompl P)
ht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)
this : ↑t * (↑sx * ↑sy * ↑sz) ∈ P
⊢ False
case this
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hu : IsUnit (x + y)
z : S
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hx : mk' S rx sx ∈ nonunits S
hrx : mk' S rx sx = x
ry : R
sy : ↥(Ideal.primeCompl P)
hy : mk' S ry sy ∈ nonunits S
hry : mk' S ry sy = y
rz : R
sz : ↥(Ideal.primeCompl P)
hxyz :
mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 { val := 1, property := (_ : 1 ∈ Ideal.primeCompl P) }
hrz : mk' S rz sz = z
t : ↥(Ideal.primeCompl P)
ht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)
⊢ ↑t * (↑sx * ↑sy * ↑sz) ∈ P | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
| exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2) | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case this
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hu : IsUnit (x + y)
z : S
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hx : mk' S rx sx ∈ nonunits S
hrx : mk' S rx sx = x
ry : R
sy : ↥(Ideal.primeCompl P)
hy : mk' S ry sy ∈ nonunits S
hry : mk' S ry sy = y
rz : R
sz : ↥(Ideal.primeCompl P)
hxyz :
mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 { val := 1, property := (_ : 1 ∈ Ideal.primeCompl P) }
hrz : mk' S rz sz = z
t : ↥(Ideal.primeCompl P)
ht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)
⊢ ↑t * (↑sx * ↑sy * ↑sz) ∈ P | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
| rw [← ht] | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
case this
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P✝ : Type u_3
inst✝¹ : CommSemiring P✝
P : Ideal R
hp : Ideal.IsPrime P
inst✝ : IsLocalization.AtPrime S P
this✝ : _root_.Nontrivial S := Nontrivial S P
x y : S
hu : IsUnit (x + y)
z : S
this : ∀ {r : R} {s : ↥(Ideal.primeCompl P)}, mk' S r s ∈ nonunits S → r ∈ P
rx : R
sx : ↥(Ideal.primeCompl P)
hx : mk' S rx sx ∈ nonunits S
hrx : mk' S rx sx = x
ry : R
sy : ↥(Ideal.primeCompl P)
hy : mk' S ry sy ∈ nonunits S
hry : mk' S ry sy = y
rz : R
sz : ↥(Ideal.primeCompl P)
hxyz :
mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 { val := 1, property := (_ : 1 ∈ Ideal.primeCompl P) }
hrz : mk' S rz sz = z
t : ↥(Ideal.primeCompl P)
ht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)
⊢ ↑t * ((rx * ↑sy + ry * ↑sx) * rz) ∈ P | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
| exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
| Mathlib.RingTheory.Localization.AtPrime.80_0.QSwWrbtcZl7L7lq | theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁶ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
P : Type u_3
inst✝³ : CommSemiring P
A : Type u_4
inst✝² : CommRing A
inst✝¹ : IsDomain A
I : Ideal R
hI : Ideal.IsPrime I
inst✝ : IsLocalization.AtPrime S I
x : R
h : optParam (LocalRing S) (_ : LocalRing S)
⊢ (algebraMap R S) x ∉ LocalRing.maximalIdeal S ↔ x ∉ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
| simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x | theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
| Mathlib.RingTheory.Localization.AtPrime.145_0.QSwWrbtcZl7L7lq | theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁶ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
P : Type u_3
inst✝³ : CommSemiring P
A : Type u_4
inst✝² : CommRing A
inst✝¹ : IsDomain A
I : Ideal R
hI : Ideal.IsPrime I
inst✝ : IsLocalization.AtPrime S I
h : optParam (LocalRing S) (_ : LocalRing S)
x : R
⊢ x ∈ Ideal.comap (algebraMap R S) (LocalRing.maximalIdeal S) ↔ x ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by | simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x | theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by | Mathlib.RingTheory.Localization.AtPrime.152_0.QSwWrbtcZl7L7lq | theorem comap_maximalIdeal (h : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁶ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
P : Type u_3
inst✝³ : CommSemiring P
A : Type u_4
inst✝² : CommRing A
inst✝¹ : IsDomain A
I : Ideal R
hI : Ideal.IsPrime I
inst✝ : IsLocalization.AtPrime S I
x : R
y : ↥(Ideal.primeCompl I)
h : optParam (LocalRing S) (_ : LocalRing S)
⊢ mk' S x y ∉ LocalRing.maximalIdeal S ↔ x ∉ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
#align is_localization.at_prime.comap_maximal_ideal IsLocalization.AtPrime.comap_maximalIdeal
theorem isUnit_mk'_iff (x : R) (y : I.primeCompl) : IsUnit (mk' S x y) ↔ x ∈ I.primeCompl :=
⟨fun h hx => mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h, fun h =>
isUnit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩
#align is_localization.at_prime.is_unit_mk'_iff IsLocalization.AtPrime.isUnit_mk'_iff
theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
| simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y | theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
| Mathlib.RingTheory.Localization.AtPrime.162_0.QSwWrbtcZl7L7lq | theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
⊢ Ideal.map (algebraMap R (Localization.AtPrime I)) I = LocalRing.maximalIdeal (Localization (Ideal.primeCompl I)) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
#align is_localization.at_prime.comap_maximal_ideal IsLocalization.AtPrime.comap_maximalIdeal
theorem isUnit_mk'_iff (x : R) (y : I.primeCompl) : IsUnit (mk' S x y) ↔ x ∈ I.primeCompl :=
⟨fun h hx => mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h, fun h =>
isUnit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩
#align is_localization.at_prime.is_unit_mk'_iff IsLocalization.AtPrime.isUnit_mk'_iff
theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y
#align is_localization.at_prime.mk'_mem_maximal_iff IsLocalization.AtPrime.mk'_mem_maximal_iff
end AtPrime
end IsLocalization
namespace Localization
open IsLocalization
attribute [local instance] Classical.propDecidable
variable (I : Ideal R) [hI : I.IsPrime]
variable {I}
/-- The unique maximal ideal of the localization at `I.primeCompl` lies over the ideal `I`. -/
theorem AtPrime.comap_maximalIdeal :
Ideal.comap (algebraMap R (Localization.AtPrime I))
(LocalRing.maximalIdeal (Localization I.primeCompl)) =
I :=
-- Porting Note : need to provide full name
IsLocalization.AtPrime.comap_maximalIdeal _ _
#align localization.at_prime.comap_maximal_ideal Localization.AtPrime.comap_maximalIdeal
/-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
| convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm | /-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
| Mathlib.RingTheory.Localization.AtPrime.192_0.QSwWrbtcZl7L7lq | /-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) | Mathlib_RingTheory_Localization_AtPrime |
case h.e'_3
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
⊢ LocalRing.maximalIdeal (Localization (Ideal.primeCompl I)) =
Ideal.map (algebraMap R (Localization.AtPrime I))
(Ideal.comap (algebraMap R (Localization.AtPrime I)) (LocalRing.maximalIdeal (Localization (Ideal.primeCompl I)))) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
#align is_localization.at_prime.comap_maximal_ideal IsLocalization.AtPrime.comap_maximalIdeal
theorem isUnit_mk'_iff (x : R) (y : I.primeCompl) : IsUnit (mk' S x y) ↔ x ∈ I.primeCompl :=
⟨fun h hx => mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h, fun h =>
isUnit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩
#align is_localization.at_prime.is_unit_mk'_iff IsLocalization.AtPrime.isUnit_mk'_iff
theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y
#align is_localization.at_prime.mk'_mem_maximal_iff IsLocalization.AtPrime.mk'_mem_maximal_iff
end AtPrime
end IsLocalization
namespace Localization
open IsLocalization
attribute [local instance] Classical.propDecidable
variable (I : Ideal R) [hI : I.IsPrime]
variable {I}
/-- The unique maximal ideal of the localization at `I.primeCompl` lies over the ideal `I`. -/
theorem AtPrime.comap_maximalIdeal :
Ideal.comap (algebraMap R (Localization.AtPrime I))
(LocalRing.maximalIdeal (Localization I.primeCompl)) =
I :=
-- Porting Note : need to provide full name
IsLocalization.AtPrime.comap_maximalIdeal _ _
#align localization.at_prime.comap_maximal_ideal Localization.AtPrime.comap_maximalIdeal
/-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm
-- Porting Note : can not find `hI`
| rw [map_comap I.primeCompl] | /-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm
-- Porting Note : can not find `hI`
| Mathlib.RingTheory.Localization.AtPrime.192_0.QSwWrbtcZl7L7lq | /-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
h : Ideal.primeCompl I ≤ Submonoid.comap f (Ideal.primeCompl J)
x : R
hx : x ∈ Ideal.comap f J
⊢ x ∈ I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
#align is_localization.at_prime.comap_maximal_ideal IsLocalization.AtPrime.comap_maximalIdeal
theorem isUnit_mk'_iff (x : R) (y : I.primeCompl) : IsUnit (mk' S x y) ↔ x ∈ I.primeCompl :=
⟨fun h hx => mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h, fun h =>
isUnit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩
#align is_localization.at_prime.is_unit_mk'_iff IsLocalization.AtPrime.isUnit_mk'_iff
theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y
#align is_localization.at_prime.mk'_mem_maximal_iff IsLocalization.AtPrime.mk'_mem_maximal_iff
end AtPrime
end IsLocalization
namespace Localization
open IsLocalization
attribute [local instance] Classical.propDecidable
variable (I : Ideal R) [hI : I.IsPrime]
variable {I}
/-- The unique maximal ideal of the localization at `I.primeCompl` lies over the ideal `I`. -/
theorem AtPrime.comap_maximalIdeal :
Ideal.comap (algebraMap R (Localization.AtPrime I))
(LocalRing.maximalIdeal (Localization I.primeCompl)) =
I :=
-- Porting Note : need to provide full name
IsLocalization.AtPrime.comap_maximalIdeal _ _
#align localization.at_prime.comap_maximal_ideal Localization.AtPrime.comap_maximalIdeal
/-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm
-- Porting Note : can not find `hI`
rw [map_comap I.primeCompl]
#align localization.at_prime.map_eq_maximal_ideal Localization.AtPrime.map_eq_maximalIdeal
theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
| contrapose! hx | theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
| Mathlib.RingTheory.Localization.AtPrime.204_0.QSwWrbtcZl7L7lq | theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
h : Ideal.primeCompl I ≤ Submonoid.comap f (Ideal.primeCompl J)
x : R
hx : x ∉ I
⊢ x ∉ Ideal.comap f J | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
#align is_localization.at_prime.comap_maximal_ideal IsLocalization.AtPrime.comap_maximalIdeal
theorem isUnit_mk'_iff (x : R) (y : I.primeCompl) : IsUnit (mk' S x y) ↔ x ∈ I.primeCompl :=
⟨fun h hx => mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h, fun h =>
isUnit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩
#align is_localization.at_prime.is_unit_mk'_iff IsLocalization.AtPrime.isUnit_mk'_iff
theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y
#align is_localization.at_prime.mk'_mem_maximal_iff IsLocalization.AtPrime.mk'_mem_maximal_iff
end AtPrime
end IsLocalization
namespace Localization
open IsLocalization
attribute [local instance] Classical.propDecidable
variable (I : Ideal R) [hI : I.IsPrime]
variable {I}
/-- The unique maximal ideal of the localization at `I.primeCompl` lies over the ideal `I`. -/
theorem AtPrime.comap_maximalIdeal :
Ideal.comap (algebraMap R (Localization.AtPrime I))
(LocalRing.maximalIdeal (Localization I.primeCompl)) =
I :=
-- Porting Note : need to provide full name
IsLocalization.AtPrime.comap_maximalIdeal _ _
#align localization.at_prime.comap_maximal_ideal Localization.AtPrime.comap_maximalIdeal
/-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm
-- Porting Note : can not find `hI`
rw [map_comap I.primeCompl]
#align localization.at_prime.map_eq_maximal_ideal Localization.AtPrime.map_eq_maximalIdeal
theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
contrapose! hx
| exact h hx | theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
contrapose! hx
| Mathlib.RingTheory.Localization.AtPrime.204_0.QSwWrbtcZl7L7lq | theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
hIJ : I = Ideal.comap f J
x : Localization.AtPrime I
hx : IsUnit ((localRingHom I J f hIJ) x)
⊢ IsUnit x | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
#align is_localization.at_prime.comap_maximal_ideal IsLocalization.AtPrime.comap_maximalIdeal
theorem isUnit_mk'_iff (x : R) (y : I.primeCompl) : IsUnit (mk' S x y) ↔ x ∈ I.primeCompl :=
⟨fun h hx => mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h, fun h =>
isUnit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩
#align is_localization.at_prime.is_unit_mk'_iff IsLocalization.AtPrime.isUnit_mk'_iff
theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y
#align is_localization.at_prime.mk'_mem_maximal_iff IsLocalization.AtPrime.mk'_mem_maximal_iff
end AtPrime
end IsLocalization
namespace Localization
open IsLocalization
attribute [local instance] Classical.propDecidable
variable (I : Ideal R) [hI : I.IsPrime]
variable {I}
/-- The unique maximal ideal of the localization at `I.primeCompl` lies over the ideal `I`. -/
theorem AtPrime.comap_maximalIdeal :
Ideal.comap (algebraMap R (Localization.AtPrime I))
(LocalRing.maximalIdeal (Localization I.primeCompl)) =
I :=
-- Porting Note : need to provide full name
IsLocalization.AtPrime.comap_maximalIdeal _ _
#align localization.at_prime.comap_maximal_ideal Localization.AtPrime.comap_maximalIdeal
/-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm
-- Porting Note : can not find `hI`
rw [map_comap I.primeCompl]
#align localization.at_prime.map_eq_maximal_ideal Localization.AtPrime.map_eq_maximalIdeal
theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
contrapose! hx
exact h hx,
fun h x hx hfxJ => hx (h hfxJ)⟩
#align localization.le_comap_prime_compl_iff Localization.le_comap_primeCompl_iff
variable (I)
/-- For a ring hom `f : R →+* S` and a prime ideal `J` in `S`, the induced ring hom from the
localization of `R` at `J.comap f` to the localization of `S` at `J`.
To make this definition more flexible, we allow any ideal `I` of `R` as input, together with a proof
that `I = J.comap f`. This can be useful when `I` is not definitionally equal to `J.comap f`.
-/
noncomputable def localRingHom (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) :
Localization.AtPrime I →+* Localization.AtPrime J :=
IsLocalization.map (Localization.AtPrime J) f (le_comap_primeCompl_iff.mpr (ge_of_eq hIJ))
#align localization.local_ring_hom Localization.localRingHom
theorem localRingHom_to_map (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f)
(x : R) : localRingHom I J f hIJ (algebraMap _ _ x) = algebraMap _ _ (f x) :=
map_eq _ _
#align localization.local_ring_hom_to_map Localization.localRingHom_to_map
theorem localRingHom_mk' (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) (x : R)
(y : I.primeCompl) :
localRingHom I J f hIJ (IsLocalization.mk' _ x y) =
IsLocalization.mk' (Localization.AtPrime J) (f x)
(⟨f y, le_comap_primeCompl_iff.mpr (ge_of_eq hIJ) y.2⟩ : J.primeCompl) :=
map_mk' _ _ _
#align localization.local_ring_hom_mk' Localization.localRingHom_mk'
instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
| rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩ | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
| Mathlib.RingTheory.Localization.AtPrime.238_0.QSwWrbtcZl7L7lq | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
hIJ : I = Ideal.comap f J
r : R
s : ↥(Ideal.primeCompl I)
hx : IsUnit ((localRingHom I J f hIJ) (mk' (Localization.AtPrime I) r s))
⊢ IsUnit (mk' (Localization.AtPrime I) r s) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
#align is_localization.at_prime.comap_maximal_ideal IsLocalization.AtPrime.comap_maximalIdeal
theorem isUnit_mk'_iff (x : R) (y : I.primeCompl) : IsUnit (mk' S x y) ↔ x ∈ I.primeCompl :=
⟨fun h hx => mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h, fun h =>
isUnit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩
#align is_localization.at_prime.is_unit_mk'_iff IsLocalization.AtPrime.isUnit_mk'_iff
theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y
#align is_localization.at_prime.mk'_mem_maximal_iff IsLocalization.AtPrime.mk'_mem_maximal_iff
end AtPrime
end IsLocalization
namespace Localization
open IsLocalization
attribute [local instance] Classical.propDecidable
variable (I : Ideal R) [hI : I.IsPrime]
variable {I}
/-- The unique maximal ideal of the localization at `I.primeCompl` lies over the ideal `I`. -/
theorem AtPrime.comap_maximalIdeal :
Ideal.comap (algebraMap R (Localization.AtPrime I))
(LocalRing.maximalIdeal (Localization I.primeCompl)) =
I :=
-- Porting Note : need to provide full name
IsLocalization.AtPrime.comap_maximalIdeal _ _
#align localization.at_prime.comap_maximal_ideal Localization.AtPrime.comap_maximalIdeal
/-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm
-- Porting Note : can not find `hI`
rw [map_comap I.primeCompl]
#align localization.at_prime.map_eq_maximal_ideal Localization.AtPrime.map_eq_maximalIdeal
theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
contrapose! hx
exact h hx,
fun h x hx hfxJ => hx (h hfxJ)⟩
#align localization.le_comap_prime_compl_iff Localization.le_comap_primeCompl_iff
variable (I)
/-- For a ring hom `f : R →+* S` and a prime ideal `J` in `S`, the induced ring hom from the
localization of `R` at `J.comap f` to the localization of `S` at `J`.
To make this definition more flexible, we allow any ideal `I` of `R` as input, together with a proof
that `I = J.comap f`. This can be useful when `I` is not definitionally equal to `J.comap f`.
-/
noncomputable def localRingHom (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) :
Localization.AtPrime I →+* Localization.AtPrime J :=
IsLocalization.map (Localization.AtPrime J) f (le_comap_primeCompl_iff.mpr (ge_of_eq hIJ))
#align localization.local_ring_hom Localization.localRingHom
theorem localRingHom_to_map (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f)
(x : R) : localRingHom I J f hIJ (algebraMap _ _ x) = algebraMap _ _ (f x) :=
map_eq _ _
#align localization.local_ring_hom_to_map Localization.localRingHom_to_map
theorem localRingHom_mk' (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) (x : R)
(y : I.primeCompl) :
localRingHom I J f hIJ (IsLocalization.mk' _ x y) =
IsLocalization.mk' (Localization.AtPrime J) (f x)
(⟨f y, le_comap_primeCompl_iff.mpr (ge_of_eq hIJ) y.2⟩ : J.primeCompl) :=
map_mk' _ _ _
#align localization.local_ring_hom_mk' Localization.localRingHom_mk'
instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩
| rw [localRingHom_mk'] at hx | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩
| Mathlib.RingTheory.Localization.AtPrime.238_0.QSwWrbtcZl7L7lq | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
hIJ : I = Ideal.comap f J
r : R
s : ↥(Ideal.primeCompl I)
hx :
IsUnit
(mk' (Localization.AtPrime J) (f r) { val := f ↑s, property := (_ : ↑s ∈ Submonoid.comap f (Ideal.primeCompl J)) })
⊢ IsUnit (mk' (Localization.AtPrime I) r s) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
#align is_localization.at_prime.comap_maximal_ideal IsLocalization.AtPrime.comap_maximalIdeal
theorem isUnit_mk'_iff (x : R) (y : I.primeCompl) : IsUnit (mk' S x y) ↔ x ∈ I.primeCompl :=
⟨fun h hx => mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h, fun h =>
isUnit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩
#align is_localization.at_prime.is_unit_mk'_iff IsLocalization.AtPrime.isUnit_mk'_iff
theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y
#align is_localization.at_prime.mk'_mem_maximal_iff IsLocalization.AtPrime.mk'_mem_maximal_iff
end AtPrime
end IsLocalization
namespace Localization
open IsLocalization
attribute [local instance] Classical.propDecidable
variable (I : Ideal R) [hI : I.IsPrime]
variable {I}
/-- The unique maximal ideal of the localization at `I.primeCompl` lies over the ideal `I`. -/
theorem AtPrime.comap_maximalIdeal :
Ideal.comap (algebraMap R (Localization.AtPrime I))
(LocalRing.maximalIdeal (Localization I.primeCompl)) =
I :=
-- Porting Note : need to provide full name
IsLocalization.AtPrime.comap_maximalIdeal _ _
#align localization.at_prime.comap_maximal_ideal Localization.AtPrime.comap_maximalIdeal
/-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm
-- Porting Note : can not find `hI`
rw [map_comap I.primeCompl]
#align localization.at_prime.map_eq_maximal_ideal Localization.AtPrime.map_eq_maximalIdeal
theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
contrapose! hx
exact h hx,
fun h x hx hfxJ => hx (h hfxJ)⟩
#align localization.le_comap_prime_compl_iff Localization.le_comap_primeCompl_iff
variable (I)
/-- For a ring hom `f : R →+* S` and a prime ideal `J` in `S`, the induced ring hom from the
localization of `R` at `J.comap f` to the localization of `S` at `J`.
To make this definition more flexible, we allow any ideal `I` of `R` as input, together with a proof
that `I = J.comap f`. This can be useful when `I` is not definitionally equal to `J.comap f`.
-/
noncomputable def localRingHom (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) :
Localization.AtPrime I →+* Localization.AtPrime J :=
IsLocalization.map (Localization.AtPrime J) f (le_comap_primeCompl_iff.mpr (ge_of_eq hIJ))
#align localization.local_ring_hom Localization.localRingHom
theorem localRingHom_to_map (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f)
(x : R) : localRingHom I J f hIJ (algebraMap _ _ x) = algebraMap _ _ (f x) :=
map_eq _ _
#align localization.local_ring_hom_to_map Localization.localRingHom_to_map
theorem localRingHom_mk' (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) (x : R)
(y : I.primeCompl) :
localRingHom I J f hIJ (IsLocalization.mk' _ x y) =
IsLocalization.mk' (Localization.AtPrime J) (f x)
(⟨f y, le_comap_primeCompl_iff.mpr (ge_of_eq hIJ) y.2⟩ : J.primeCompl) :=
map_mk' _ _ _
#align localization.local_ring_hom_mk' Localization.localRingHom_mk'
instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩
rw [localRingHom_mk'] at hx
| rw [AtPrime.isUnit_mk'_iff] at hx ⊢ | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩
rw [localRingHom_mk'] at hx
| Mathlib.RingTheory.Localization.AtPrime.238_0.QSwWrbtcZl7L7lq | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) | Mathlib_RingTheory_Localization_AtPrime |
case intro.intro
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
P : Type u_3
inst✝ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
J : Ideal P
hJ : Ideal.IsPrime J
f : R →+* P
hIJ : I = Ideal.comap f J
r : R
s : ↥(Ideal.primeCompl I)
hx : f r ∈ Ideal.primeCompl J
⊢ r ∈ Ideal.primeCompl I | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
#align is_localization.at_prime.comap_maximal_ideal IsLocalization.AtPrime.comap_maximalIdeal
theorem isUnit_mk'_iff (x : R) (y : I.primeCompl) : IsUnit (mk' S x y) ↔ x ∈ I.primeCompl :=
⟨fun h hx => mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h, fun h =>
isUnit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩
#align is_localization.at_prime.is_unit_mk'_iff IsLocalization.AtPrime.isUnit_mk'_iff
theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y
#align is_localization.at_prime.mk'_mem_maximal_iff IsLocalization.AtPrime.mk'_mem_maximal_iff
end AtPrime
end IsLocalization
namespace Localization
open IsLocalization
attribute [local instance] Classical.propDecidable
variable (I : Ideal R) [hI : I.IsPrime]
variable {I}
/-- The unique maximal ideal of the localization at `I.primeCompl` lies over the ideal `I`. -/
theorem AtPrime.comap_maximalIdeal :
Ideal.comap (algebraMap R (Localization.AtPrime I))
(LocalRing.maximalIdeal (Localization I.primeCompl)) =
I :=
-- Porting Note : need to provide full name
IsLocalization.AtPrime.comap_maximalIdeal _ _
#align localization.at_prime.comap_maximal_ideal Localization.AtPrime.comap_maximalIdeal
/-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm
-- Porting Note : can not find `hI`
rw [map_comap I.primeCompl]
#align localization.at_prime.map_eq_maximal_ideal Localization.AtPrime.map_eq_maximalIdeal
theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
contrapose! hx
exact h hx,
fun h x hx hfxJ => hx (h hfxJ)⟩
#align localization.le_comap_prime_compl_iff Localization.le_comap_primeCompl_iff
variable (I)
/-- For a ring hom `f : R →+* S` and a prime ideal `J` in `S`, the induced ring hom from the
localization of `R` at `J.comap f` to the localization of `S` at `J`.
To make this definition more flexible, we allow any ideal `I` of `R` as input, together with a proof
that `I = J.comap f`. This can be useful when `I` is not definitionally equal to `J.comap f`.
-/
noncomputable def localRingHom (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) :
Localization.AtPrime I →+* Localization.AtPrime J :=
IsLocalization.map (Localization.AtPrime J) f (le_comap_primeCompl_iff.mpr (ge_of_eq hIJ))
#align localization.local_ring_hom Localization.localRingHom
theorem localRingHom_to_map (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f)
(x : R) : localRingHom I J f hIJ (algebraMap _ _ x) = algebraMap _ _ (f x) :=
map_eq _ _
#align localization.local_ring_hom_to_map Localization.localRingHom_to_map
theorem localRingHom_mk' (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) (x : R)
(y : I.primeCompl) :
localRingHom I J f hIJ (IsLocalization.mk' _ x y) =
IsLocalization.mk' (Localization.AtPrime J) (f x)
(⟨f y, le_comap_primeCompl_iff.mpr (ge_of_eq hIJ) y.2⟩ : J.primeCompl) :=
map_mk' _ _ _
#align localization.local_ring_hom_mk' Localization.localRingHom_mk'
instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩
rw [localRingHom_mk'] at hx
rw [AtPrime.isUnit_mk'_iff] at hx ⊢
| exact fun hr => hx ((SetLike.ext_iff.mp hIJ r).mp hr) | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩
rw [localRingHom_mk'] at hx
rw [AtPrime.isUnit_mk'_iff] at hx ⊢
| Mathlib.RingTheory.Localization.AtPrime.238_0.QSwWrbtcZl7L7lq | instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S✝ : Type u_2
inst✝³ : CommSemiring S✝
inst✝² : Algebra R S✝
P : Type u_3
inst✝¹ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
S : Type u_4
inst✝ : CommSemiring S
J : Ideal S
hJ : Ideal.IsPrime J
K : Ideal P
hK : Ideal.IsPrime K
f : R →+* S
hIJ : I = Ideal.comap f J
g : S →+* P
hJK : J = Ideal.comap g K
⊢ I = Ideal.comap (RingHom.comp g f) K | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
#align is_localization.at_prime.comap_maximal_ideal IsLocalization.AtPrime.comap_maximalIdeal
theorem isUnit_mk'_iff (x : R) (y : I.primeCompl) : IsUnit (mk' S x y) ↔ x ∈ I.primeCompl :=
⟨fun h hx => mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h, fun h =>
isUnit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩
#align is_localization.at_prime.is_unit_mk'_iff IsLocalization.AtPrime.isUnit_mk'_iff
theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y
#align is_localization.at_prime.mk'_mem_maximal_iff IsLocalization.AtPrime.mk'_mem_maximal_iff
end AtPrime
end IsLocalization
namespace Localization
open IsLocalization
attribute [local instance] Classical.propDecidable
variable (I : Ideal R) [hI : I.IsPrime]
variable {I}
/-- The unique maximal ideal of the localization at `I.primeCompl` lies over the ideal `I`. -/
theorem AtPrime.comap_maximalIdeal :
Ideal.comap (algebraMap R (Localization.AtPrime I))
(LocalRing.maximalIdeal (Localization I.primeCompl)) =
I :=
-- Porting Note : need to provide full name
IsLocalization.AtPrime.comap_maximalIdeal _ _
#align localization.at_prime.comap_maximal_ideal Localization.AtPrime.comap_maximalIdeal
/-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm
-- Porting Note : can not find `hI`
rw [map_comap I.primeCompl]
#align localization.at_prime.map_eq_maximal_ideal Localization.AtPrime.map_eq_maximalIdeal
theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
contrapose! hx
exact h hx,
fun h x hx hfxJ => hx (h hfxJ)⟩
#align localization.le_comap_prime_compl_iff Localization.le_comap_primeCompl_iff
variable (I)
/-- For a ring hom `f : R →+* S` and a prime ideal `J` in `S`, the induced ring hom from the
localization of `R` at `J.comap f` to the localization of `S` at `J`.
To make this definition more flexible, we allow any ideal `I` of `R` as input, together with a proof
that `I = J.comap f`. This can be useful when `I` is not definitionally equal to `J.comap f`.
-/
noncomputable def localRingHom (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) :
Localization.AtPrime I →+* Localization.AtPrime J :=
IsLocalization.map (Localization.AtPrime J) f (le_comap_primeCompl_iff.mpr (ge_of_eq hIJ))
#align localization.local_ring_hom Localization.localRingHom
theorem localRingHom_to_map (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f)
(x : R) : localRingHom I J f hIJ (algebraMap _ _ x) = algebraMap _ _ (f x) :=
map_eq _ _
#align localization.local_ring_hom_to_map Localization.localRingHom_to_map
theorem localRingHom_mk' (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) (x : R)
(y : I.primeCompl) :
localRingHom I J f hIJ (IsLocalization.mk' _ x y) =
IsLocalization.mk' (Localization.AtPrime J) (f x)
(⟨f y, le_comap_primeCompl_iff.mpr (ge_of_eq hIJ) y.2⟩ : J.primeCompl) :=
map_mk' _ _ _
#align localization.local_ring_hom_mk' Localization.localRingHom_mk'
instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩
rw [localRingHom_mk'] at hx
rw [AtPrime.isUnit_mk'_iff] at hx ⊢
exact fun hr => hx ((SetLike.ext_iff.mp hIJ r).mp hr)
#align localization.is_local_ring_hom_local_ring_hom Localization.isLocalRingHom_localRingHom
theorem localRingHom_unique (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f)
{j : Localization.AtPrime I →+* Localization.AtPrime J}
(hj : ∀ x : R, j (algebraMap _ _ x) = algebraMap _ _ (f x)) : localRingHom I J f hIJ = j :=
map_unique _ _ hj
#align localization.local_ring_hom_unique Localization.localRingHom_unique
@[simp]
theorem localRingHom_id : localRingHom I I (RingHom.id R) (Ideal.comap_id I).symm = RingHom.id _ :=
localRingHom_unique _ _ _ _ fun _ => rfl
#align localization.local_ring_hom_id Localization.localRingHom_id
-- Porting note : simplifier won't pick up this lemma, so deleted @[simp]
theorem localRingHom_comp {S : Type*} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P)
[hK : K.IsPrime] (f : R →+* S) (hIJ : I = J.comap f) (g : S →+* P) (hJK : J = K.comap g) :
localRingHom I K (g.comp f) (by | rw [hIJ, hJK, Ideal.comap_comap f g] | theorem localRingHom_comp {S : Type*} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P)
[hK : K.IsPrime] (f : R →+* S) (hIJ : I = J.comap f) (g : S →+* P) (hJK : J = K.comap g) :
localRingHom I K (g.comp f) (by | Mathlib.RingTheory.Localization.AtPrime.259_0.QSwWrbtcZl7L7lq | theorem localRingHom_comp {S : Type*} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P)
[hK : K.IsPrime] (f : R →+* S) (hIJ : I = J.comap f) (g : S →+* P) (hJK : J = K.comap g) :
localRingHom I K (g.comp f) (by rw [hIJ, hJK, Ideal.comap_comap f g]) =
(localRingHom J K g hJK).comp (localRingHom I J f hIJ) | Mathlib_RingTheory_Localization_AtPrime |
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S✝ : Type u_2
inst✝³ : CommSemiring S✝
inst✝² : Algebra R S✝
P : Type u_3
inst✝¹ : CommSemiring P
I : Ideal R
hI : Ideal.IsPrime I
S : Type u_4
inst✝ : CommSemiring S
J : Ideal S
hJ : Ideal.IsPrime J
K : Ideal P
hK : Ideal.IsPrime K
f : R →+* S
hIJ : I = Ideal.comap f J
g : S →+* P
hJK : J = Ideal.comap g K
r : R
⊢ (RingHom.comp (localRingHom J K g hJK) (localRingHom I J f hIJ)) ((algebraMap R (Localization.AtPrime I)) r) =
(algebraMap P (Localization.AtPrime K)) ((RingHom.comp g f) r) | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.Ideal
#align_import ring_theory.localization.at_prime from "leanprover-community/mathlib"@"b86c528d08a52a1fdb50d999232408e1c7e85d7d"
/-!
# Localizations of commutative rings at the complement of a prime ideal
## Main definitions
* `IsLocalization.AtPrime (P : Ideal R) [IsPrime P] (S : Type*)` expresses that `S` is a
localization at (the complement of) a prime ideal `P`, as an abbreviation of
`IsLocalization P.prime_compl S`
## Main results
* `IsLocalization.AtPrime.localRing`: a theorem (not an instance) stating a localization at the
complement of a prime ideal is a local ring
## Implementation notes
See `RingTheory.Localization.Basic` for a design overview.
## Tags
localization, ring localization, commutative ring localization, characteristic predicate,
commutative ring, field of fractions
-/
variable {R : Type*} [CommSemiring R] (M : Submonoid R) (S : Type*) [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
section AtPrime
variable (P : Ideal R) [hp : P.IsPrime]
namespace Ideal
/-- The complement of a prime ideal `P ⊆ R` is a submonoid of `R`. -/
def primeCompl : Submonoid R where
carrier := (Pᶜ : Set R)
one_mem' := by convert P.ne_top_iff_one.1 hp.1
mul_mem' {x y} hnx hny hxy := Or.casesOn (hp.mem_or_mem hxy) hnx hny
#align ideal.prime_compl Ideal.primeCompl
theorem primeCompl_le_nonZeroDivisors [NoZeroDivisors R] : P.primeCompl ≤ nonZeroDivisors R :=
le_nonZeroDivisors_of_noZeroDivisors <| not_not_intro P.zero_mem
#align ideal.prime_compl_le_non_zero_divisors Ideal.primeCompl_le_nonZeroDivisors
end Ideal
/-- Given a prime ideal `P`, the typeclass `IsLocalization.AtPrime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. -/
protected abbrev IsLocalization.AtPrime :=
IsLocalization P.primeCompl S
#align is_localization.at_prime IsLocalization.AtPrime
/-- Given a prime ideal `P`, `Localization.AtPrime P` is a localization of
`R` at the complement of `P`, as a quotient type. -/
protected abbrev Localization.AtPrime :=
Localization P.primeCompl
#align localization.at_prime Localization.AtPrime
namespace IsLocalization
theorem AtPrime.Nontrivial [IsLocalization.AtPrime S P] : Nontrivial S :=
nontrivial_of_ne (0 : S) 1 fun hze => by
rw [← (algebraMap R S).map_one, ← (algebraMap R S).map_zero] at hze
obtain ⟨t, ht⟩ := (eq_iff_exists P.primeCompl S).1 hze
have htz : (t : R) = 0 := by simpa using ht.symm
exact t.2 (htz.symm ▸ P.zero_mem : ↑t ∈ P)
#align is_localization.at_prime.nontrivial IsLocalization.AtPrime.Nontrivial
theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
-- Porting Note : since I couldn't get local instance running, I just specify it manually
letI := AtPrime.Nontrivial S P
LocalRing.of_nonunits_add
(by
intro x y hx hy hu
cases' isUnit_iff_exists_inv.1 hu with z hxyz
have : ∀ {r : R} {s : P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P := fun {r s} =>
not_imp_comm.1 fun nr => isUnit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : P.primeCompl),
mk'_mul_mk'_eq_one' _ _ <| show r ∈ P.primeCompl from nr⟩
rcases mk'_surjective P.primeCompl x with ⟨rx, sx, hrx⟩
rcases mk'_surjective P.primeCompl y with ⟨ry, sy, hry⟩
rcases mk'_surjective P.primeCompl z with ⟨rz, sz, hrz⟩
rw [← hrx, ← hry, ← hrz, ← mk'_add, ← mk'_mul, ← mk'_self S P.primeCompl.one_mem] at hxyz
rw [← hrx] at hx
rw [← hry] at hy
obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
suffices : (t : R) * (sx * sy * sz) ∈ P
exact
not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
(hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
rw [← ht]
exact
P.mul_mem_left _ <| P.mul_mem_right _ <|
P.add_mem (P.mul_mem_right _ <| this hx) <| P.mul_mem_right _ <| this hy)
#align is_localization.at_prime.local_ring IsLocalization.AtPrime.localRing
end IsLocalization
namespace Localization
/-- The localization of `R` at the complement of a prime ideal is a local ring. -/
instance AtPrime.localRing : LocalRing (Localization P.primeCompl) :=
IsLocalization.AtPrime.localRing (Localization P.primeCompl) P
#align localization.at_prime.local_ring Localization.AtPrime.localRing
end Localization
end AtPrime
namespace IsLocalization
variable {A : Type*} [CommRing A] [IsDomain A]
/-- The localization of an integral domain at the complement of a prime ideal is an integral domain.
-/
instance isDomain_of_local_atPrime {P : Ideal A} (_ : P.IsPrime) :
IsDomain (Localization.AtPrime P) :=
isDomain_localization P.primeCompl_le_nonZeroDivisors
#align is_localization.is_domain_of_local_at_prime IsLocalization.isDomain_of_local_atPrime
namespace AtPrime
variable (I : Ideal R) [hI : I.IsPrime] [IsLocalization.AtPrime S I]
theorem isUnit_to_map_iff (x : R) : IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl :=
⟨fun h hx =>
(isPrime_of_isPrime_disjoint I.primeCompl S I hI disjoint_compl_left).ne_top <|
(Ideal.map (algebraMap R S) I).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ hx) h,
fun h => map_units S ⟨x, h⟩⟩
#align is_localization.at_prime.is_unit_to_map_iff IsLocalization.AtPrime.isUnit_to_map_iff
-- Can't use typeclasses to infer the `LocalRing` instance, so use an `optParam` instead
-- (since `LocalRing` is a `Prop`, there should be no unification issues.)
theorem to_map_mem_maximal_iff (x : R) (h : LocalRing S := localRing S I) :
algebraMap R S x ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_to_map_iff S I x
#align is_localization.at_prime.to_map_mem_maximal_iff IsLocalization.AtPrime.to_map_mem_maximal_iff
theorem comap_maximalIdeal (h : LocalRing S := localRing S I) :
(LocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
#align is_localization.at_prime.comap_maximal_ideal IsLocalization.AtPrime.comap_maximalIdeal
theorem isUnit_mk'_iff (x : R) (y : I.primeCompl) : IsUnit (mk' S x y) ↔ x ∈ I.primeCompl :=
⟨fun h hx => mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h, fun h =>
isUnit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩
#align is_localization.at_prime.is_unit_mk'_iff IsLocalization.AtPrime.isUnit_mk'_iff
theorem mk'_mem_maximal_iff (x : R) (y : I.primeCompl) (h : LocalRing S := localRing S I) :
mk' S x y ∈ LocalRing.maximalIdeal S ↔ x ∈ I :=
not_iff_not.mp <| by
simpa only [LocalRing.mem_maximalIdeal, mem_nonunits_iff, Classical.not_not] using
isUnit_mk'_iff S I x y
#align is_localization.at_prime.mk'_mem_maximal_iff IsLocalization.AtPrime.mk'_mem_maximal_iff
end AtPrime
end IsLocalization
namespace Localization
open IsLocalization
attribute [local instance] Classical.propDecidable
variable (I : Ideal R) [hI : I.IsPrime]
variable {I}
/-- The unique maximal ideal of the localization at `I.primeCompl` lies over the ideal `I`. -/
theorem AtPrime.comap_maximalIdeal :
Ideal.comap (algebraMap R (Localization.AtPrime I))
(LocalRing.maximalIdeal (Localization I.primeCompl)) =
I :=
-- Porting Note : need to provide full name
IsLocalization.AtPrime.comap_maximalIdeal _ _
#align localization.at_prime.comap_maximal_ideal Localization.AtPrime.comap_maximalIdeal
/-- The image of `I` in the localization at `I.primeCompl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `AtPrime.localRing` -/
theorem AtPrime.map_eq_maximalIdeal :
Ideal.map (algebraMap R (Localization.AtPrime I)) I =
LocalRing.maximalIdeal (Localization I.primeCompl) := by
convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))
-- Porting Note : `algebraMap R ...` can not be solve by unification
(AtPrime.comap_maximalIdeal (hI := hI)).symm
-- Porting Note : can not find `hI`
rw [map_comap I.primeCompl]
#align localization.at_prime.map_eq_maximal_ideal Localization.AtPrime.map_eq_maximalIdeal
theorem le_comap_primeCompl_iff {J : Ideal P} [hJ : J.IsPrime] {f : R →+* P} :
I.primeCompl ≤ J.primeCompl.comap f ↔ J.comap f ≤ I :=
⟨fun h x hx => by
contrapose! hx
exact h hx,
fun h x hx hfxJ => hx (h hfxJ)⟩
#align localization.le_comap_prime_compl_iff Localization.le_comap_primeCompl_iff
variable (I)
/-- For a ring hom `f : R →+* S` and a prime ideal `J` in `S`, the induced ring hom from the
localization of `R` at `J.comap f` to the localization of `S` at `J`.
To make this definition more flexible, we allow any ideal `I` of `R` as input, together with a proof
that `I = J.comap f`. This can be useful when `I` is not definitionally equal to `J.comap f`.
-/
noncomputable def localRingHom (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) :
Localization.AtPrime I →+* Localization.AtPrime J :=
IsLocalization.map (Localization.AtPrime J) f (le_comap_primeCompl_iff.mpr (ge_of_eq hIJ))
#align localization.local_ring_hom Localization.localRingHom
theorem localRingHom_to_map (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f)
(x : R) : localRingHom I J f hIJ (algebraMap _ _ x) = algebraMap _ _ (f x) :=
map_eq _ _
#align localization.local_ring_hom_to_map Localization.localRingHom_to_map
theorem localRingHom_mk' (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f) (x : R)
(y : I.primeCompl) :
localRingHom I J f hIJ (IsLocalization.mk' _ x y) =
IsLocalization.mk' (Localization.AtPrime J) (f x)
(⟨f y, le_comap_primeCompl_iff.mpr (ge_of_eq hIJ) y.2⟩ : J.primeCompl) :=
map_mk' _ _ _
#align localization.local_ring_hom_mk' Localization.localRingHom_mk'
instance isLocalRingHom_localRingHom (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P)
(hIJ : I = J.comap f) : IsLocalRingHom (localRingHom I J f hIJ) :=
IsLocalRingHom.mk fun x hx => by
rcases IsLocalization.mk'_surjective I.primeCompl x with ⟨r, s, rfl⟩
rw [localRingHom_mk'] at hx
rw [AtPrime.isUnit_mk'_iff] at hx ⊢
exact fun hr => hx ((SetLike.ext_iff.mp hIJ r).mp hr)
#align localization.is_local_ring_hom_local_ring_hom Localization.isLocalRingHom_localRingHom
theorem localRingHom_unique (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = J.comap f)
{j : Localization.AtPrime I →+* Localization.AtPrime J}
(hj : ∀ x : R, j (algebraMap _ _ x) = algebraMap _ _ (f x)) : localRingHom I J f hIJ = j :=
map_unique _ _ hj
#align localization.local_ring_hom_unique Localization.localRingHom_unique
@[simp]
theorem localRingHom_id : localRingHom I I (RingHom.id R) (Ideal.comap_id I).symm = RingHom.id _ :=
localRingHom_unique _ _ _ _ fun _ => rfl
#align localization.local_ring_hom_id Localization.localRingHom_id
-- Porting note : simplifier won't pick up this lemma, so deleted @[simp]
theorem localRingHom_comp {S : Type*} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P)
[hK : K.IsPrime] (f : R →+* S) (hIJ : I = J.comap f) (g : S →+* P) (hJK : J = K.comap g) :
localRingHom I K (g.comp f) (by rw [hIJ, hJK, Ideal.comap_comap f g]) =
(localRingHom J K g hJK).comp (localRingHom I J f hIJ) :=
localRingHom_unique _ _ _ _ fun r => by
| simp only [Function.comp_apply, RingHom.coe_comp, localRingHom_to_map] | theorem localRingHom_comp {S : Type*} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P)
[hK : K.IsPrime] (f : R →+* S) (hIJ : I = J.comap f) (g : S →+* P) (hJK : J = K.comap g) :
localRingHom I K (g.comp f) (by rw [hIJ, hJK, Ideal.comap_comap f g]) =
(localRingHom J K g hJK).comp (localRingHom I J f hIJ) :=
localRingHom_unique _ _ _ _ fun r => by
| Mathlib.RingTheory.Localization.AtPrime.259_0.QSwWrbtcZl7L7lq | theorem localRingHom_comp {S : Type*} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P)
[hK : K.IsPrime] (f : R →+* S) (hIJ : I = J.comap f) (g : S →+* P) (hJK : J = K.comap g) :
localRingHom I K (g.comp f) (by rw [hIJ, hJK, Ideal.comap_comap f g]) =
(localRingHom J K g hJK).comp (localRingHom I J f hIJ) | Mathlib_RingTheory_Localization_AtPrime |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j : ι
h : ¬ComplexShape.Rel c i j
⊢ (fun i j => (d C i j).app t) i j = 0 | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T ⥤ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ι : Type*} {c : ComplexShape ι}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
| have := C.shape _ _ h | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j : ι
h : ¬ComplexShape.Rel c i j
this : d C i j = 0
⊢ (fun i j => (d C i j).app t) i j = 0 | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T ⥤ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ι : Type*} {c : ComplexShape ι}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
| rw [NatTrans.ext_iff, Function.funext_iff] at this | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j : ι
h : ¬ComplexShape.Rel c i j
this : ∀ (a : T), (d C i j).app a = 0.app a
⊢ (fun i j => (d C i j).app t) i j = 0 | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T ⥤ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ι : Type*} {c : ComplexShape ι}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
| exact this t | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j k : ι
x✝¹ : ComplexShape.Rel c i j
x✝ : ComplexShape.Rel c j k
⊢ (fun i j => (d C i j).app t) i j ≫ (fun i j => (d C i j).app t) j k = 0 | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T ⥤ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ι : Type*} {c : ComplexShape ι}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
| have := C.d_comp_d i j k | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j k : ι
x✝¹ : ComplexShape.Rel c i j
x✝ : ComplexShape.Rel c j k
this : d C i j ≫ d C j k = 0
⊢ (fun i j => (d C i j).app t) i j ≫ (fun i j => (d C i j).app t) j k = 0 | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T ⥤ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ι : Type*} {c : ComplexShape ι}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
| rw [NatTrans.ext_iff, Function.funext_iff] at this | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i j k : ι
x✝¹ : ComplexShape.Rel c i j
x✝ : ComplexShape.Rel c j k
this : ∀ (a : T), (d C i j ≫ d C j k).app a = 0.app a
⊢ (fun i j => (d C i j).app t) i j ≫ (fun i j => (d C i j).app t) j k = 0 | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T ⥤ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ι : Type*} {c : ComplexShape ι}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
| exact this t | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
⊢ { obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Hom.mk fun i => (HomologicalComplex.X C i).map h }.map
(𝟙 t) =
𝟙
({ obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Hom.mk fun i => (HomologicalComplex.X C i).map h }.obj
t) | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T ⥤ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ι : Type*} {c : ComplexShape ι}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
| ext i | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
case h
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i : ι
⊢ Hom.f
({ obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Hom.mk fun i => (HomologicalComplex.X C i).map h }.map
(𝟙 t))
i =
Hom.f
(𝟙
({ obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h => Hom.mk fun i => (HomologicalComplex.X C i).map h }.obj
t))
i | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T ⥤ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ι : Type*} {c : ComplexShape ι}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
| dsimp | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
case h
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
t : T
i : ι
⊢ (X C i).map (𝟙 t) = 𝟙 ((X C i).obj t) | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb84f"
/-!
# Complexes in functor categories
We can view a complex valued in a functor category `T ⥤ V` as
a functor from `T` to complexes valued in `V`.
## Future work
In fact this is an equivalence of categories.
-/
universe v u
open CategoryTheory
open CategoryTheory.Limits
namespace HomologicalComplex
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
variable {ι : Type*} {c : ComplexShape ι}
/-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
dsimp
| rw [(C.X i).map_id] | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
have := C.d_comp_d i j k
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t
shape := fun i j h => by
have := C.shape _ _ h
rw [NatTrans.ext_iff, Function.funext_iff] at this
exact this t }
map h :=
{ f := fun i => (C.X i).map h
comm' := fun i j _ => NatTrans.naturality _ _ }
map_id t := by
ext i
dsimp
| Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
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