File size: 4,507 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 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | import OAI.Algebra.Finitistic.Rectification.ColumnProperties
import Mathlib.Algebra.Category.ModuleCat.Biproducts
namespace OAI
open CategoryTheory CategoryTheory.Limits HomologicalComplex
open scoped TensorProduct ModuleCat.Algebra
noncomputable section
namespace ExternalTensor.Section4
universe u
variable (k B : Type u) [Field k] [Ring B] [Algebra k B]
attribute [local instance] ExternalTensor.bimodule ExternalTensor.rightModule
def forgetK : ModuleCat.{u} (Enveloping k B) ⥤ ModuleCat.{u} k :=
ModuleCat.restrictScalars (algebraMap k (Enveloping k B))
instance forgetK_additive : (forgetK k B).Additive := by
dsimp [forgetK]
infer_instance
def columnKIso (e : B) (N : ModuleCat.{u} Bᵐᵒᵖ) :
(forgetK k B).obj ((columnFunctor k B e).obj N) ≅
ModuleCat.of k (Column k B N e) :=
(show ((forgetK k B).obj ((columnFunctor k B e).obj N)) ≃ₗ[k]
(ModuleCat.of k (Column k B N e)) from
{ toFun := id
invFun := id
left_inv := fun _ => rfl
right_inv := fun _ => rfl
map_add' := fun _ _ => rfl
map_smul' := fun c x => by
change (algebraMap k (Enveloping k B) c) •
(show Column k B N e from x) = c • (show Column k B N e from x)
exact IsScalarTower.algebraMap_smul (Enveloping k B) c (show Column k B N e from x) }).toModuleIso
theorem finite_of_iso {X Y : ModuleCat.{u} k} (e : X ≅ Y)
(hY : FiniteDimensional k Y) : FiniteDimensional k X := by
let := hY
exact FiniteDimensional.of_injective e.hom.hom e.toLinearEquiv.injective
theorem column_finite [FiniteDimensional k B] (e : B)
(N : ModuleCat.{u} Bᵐᵒᵖ) [Module.Finite Bᵐᵒᵖ N] :
FiniteDimensional k ((forgetK k B).obj ((columnFunctor k B e).obj N)) :=
finite_of_iso k (columnKIso k B e N) (column_finiteDimensional k B N e)
abbrev termK (n : ℤ) : BiComplex k B ⥤ ModuleCat.{u} k :=
eval _ _ n ⋙ forgetK k B
theorem sum_finite {I : Type} [Fintype I] (K : I → BiComplex k B)
(n : ℤ) (hK : ∀ i, FiniteDimensional k ((termK k B n).obj (K i))) :
FiniteDimensional k ((termK k B n).obj (⨁ K)) := by
let (i : I) : FiniteDimensional k ((termK k B n).obj (K i)) := hK i
apply finite_of_iso k ((termK k B n).mapBiproduct K ≪≫ ModuleCat.biproductIsoPi _)
change Module.Finite k (∀ i, ((termK k B n).obj (K i)))
infer_instance
theorem biprod_finite {X Y : ModuleCat.{u} k}
(hX : FiniteDimensional k X) (hY : FiniteDimensional k Y) :
FiniteDimensional k (X ⊞ Y : ModuleCat.{u} k) := by
let := hX
let := hY
apply finite_of_iso k (ModuleCat.biprodIsoProd X Y)
change Module.Finite k (X × Y)
infer_instance
theorem biprod_term_finite (K L : BiComplex k B) (n : ℤ)
(hK : FiniteDimensional k ((termK k B n).obj K))
(hL : FiniteDimensional k ((termK k B n).obj L)) :
FiniteDimensional k ((termK k B n).obj (K ⊞ L)) := by
let := preservesBinaryBiproducts_of_preservesBiproducts (termK k B n)
exact finite_of_iso k ((termK k B n).mapBiprod K L) (biprod_finite k hK hL)
variable {k B}
variable {A J R : Type} [Fintype A] [Fintype J] [Fintype R]
variable (s : Quadratic k B (A := A) (J := J) (R := R))
variable (D₀ D₁ D₂ : RightComplex B)
variable (f : A → (D₀ ⟶ D₁)) (g : J → (D₁ ⟶ D₂))
/-- Finite-dimensionality of the actual three induced columns, from exactly
finite-dimensional B and finite generation of the right-module input terms. -/
theorem columns_finite [FiniteDimensional k B]
(h₀ : ∀ n, Module.Finite Bᵐᵒᵖ (D₀.X n))
(h₁ : ∀ n, Module.Finite Bᵐᵒᵖ (D₁.X n))
(h₂ : ∀ n, Module.Finite Bᵐᵒᵖ (D₂.X n)) (n : ℤ) :
FiniteDimensional k ((forgetK k B).obj ((L0Prime s D₀ D₁ D₂ f g).X n)) ∧
FiniteDimensional k ((forgetK k B).obj ((L1Prime s D₀ D₁ D₂ f g).X n)) ∧
FiniteDimensional k ((forgetK k B).obj ((L2Prime s D₀ D₁ D₂ f g).X n)) := by
let := h₀ n
let := h₁ n
let := h₂ n
refine ⟨?_, ?_, ?_⟩
· apply biprod_term_finite
· exact column_finite k B s.e₀ (D₀.X n)
· apply biprod_term_finite
· exact column_finite k B s.e₁ (D₁.X n)
· exact column_finite k B s.e₂ (D₂.X n)
· apply biprod_term_finite
· apply sum_finite
intro a
exact column_finite k B s.e₁ (D₀.X n)
· apply sum_finite
intro b
exact column_finite k B s.e₂ (D₁.X n)
· apply sum_finite
intro r
exact column_finite k B s.e₂ (D₀.X n)
end ExternalTensor.Section4
end
end OAI
|