File size: 3,321 Bytes
ae00908 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import Mathlib.LinearAlgebra.TensorProduct.Tower
import OAI.Algebra.Finitistic.Rectification.ExternalTensor
namespace OAI
noncomputable section
open scoped TensorProduct
namespace LittleFinitistic.TensorProjection
attribute [local instance] ExternalTensor.rightModule
variable {k B M M' N : Type*} [Field k] [Ring B] [Algebra k B]
[AddCommGroup M] [Module k M] [AddCommGroup M'] [Module k M']
[AddCommGroup N] [Module k N] [Module Bᵐᵒᵖ N] [IsScalarTower k Bᵐᵒᵖ N]
/-- Tensor a k-linear map with the identity while retaining the actual right B-action. -/
def rightMap (f : M →ₗ[k] M') : (M ⊗[k] N) →ₗ[Bᵐᵒᵖ] (M' ⊗[k] N) where
__ := (TensorProduct.map f LinearMap.id).toAddMonoidHom
map_smul' c x := by
induction x using TensorProduct.inductionOn with
| tmul m n => simp [ExternalTensor.right_smul_tmul]
| add x y hx hy => simp_all [smul_add]
@[simp] theorem rightMap_tmul (f : M →ₗ[k] M') (m : M) (n : N) :
rightMap (B := B) f (m ⊗ₜ[k] n) = f m ⊗ₜ[k] n := rfl
/-- Transport a first-factor vector-space equivalence as an actual right-module equivalence. -/
def rightCongr (e : M ≃ₗ[k] M') : (M ⊗[k] N) ≃ₗ[Bᵐᵒᵖ] (M' ⊗[k] N) where
__ := rightMap (B := B) (N := N) e.toLinearMap
invFun := rightMap (B := B) (N := N) e.symm.toLinearMap
left_inv x := by
induction x using TensorProduct.inductionOn with
| tmul m n => simp
| add x y hx hy => simp_all
right_inv x := by
induction x using TensorProduct.inductionOn with
| tmul m n => simp
| add x y hx hy => simp_all
variable (p : M →ₗ[k] M)
/-- The actual fixed submodule for a projector on the first tensor factor. -/
def fixed : Submodule Bᵐᵒᵖ (M ⊗[k] N) := (rightMap (B := B) p - LinearMap.id).ker
@[simp] theorem mem_fixed (x : M ⊗[k] N) : x ∈ fixed (B := B) p ↔ rightMap (B := B) p x = x := by
simp [fixed,sub_eq_zero]
theorem range_fixed (hp : p.comp p = p) (x : p.range) : p x.val = x.val := by
obtain ⟨m,hm⟩ := x.property
have h := LinearMap.congr_fun hp m
simpa only [LinearMap.comp_apply,hm] using h
/-- The eigenspace e(M tensor N) is exactly (eM) tensor N as a right module. -/
def imageTensorEquiv (hp : p.comp p = p) : (p.range ⊗[k] N) ≃ₗ[Bᵐᵒᵖ] fixed (B := B) (N := N) p where
toFun x := ⟨rightMap (B := B) (N := N) p.range.subtype x,by
rw [mem_fixed]
induction x using TensorProduct.inductionOn with
| tmul m n => simp [range_fixed p hp m]
| add x y hx hy => simp_all⟩
invFun x := rightMap (B := B) (N := N) p.rangeRestrict x.val
left_inv x := by
induction x using TensorProduct.inductionOn with
| tmul m n =>
simp only [rightMap_tmul]
congr 1
apply Subtype.ext
exact range_fixed p hp m
| add x y hx hy => simp_all
right_inv x := by
apply Subtype.ext
have h : rightMap (B := B) p.range.subtype (rightMap (B := B) (N := N) p.rangeRestrict x.val) =
rightMap (B := B) p x.val := by
generalize x.val = y
induction y using TensorProduct.inductionOn with
| tmul m n => rfl
| add a b ha hb => simp_all
exact h.trans ((mem_fixed p x.val).1 x.property)
map_add' x y := Subtype.ext (map_add _ x y)
map_smul' c x := Subtype.ext (map_smul _ c x)
end LittleFinitistic.TensorProjection
end
end OAI
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