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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import OAI.Algebra.Finitistic.Rectification.InducedColumns
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.HomologicalComplexLimits
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]
instance rightModuleK (N : ModuleCat.{u} Bᵐᵒᵖ) : Module k N :=
Module.compHom N (algebraMap k Bᵐᵒᵖ)
instance rightModuleTower (N : ModuleCat.{u} Bᵐᵒᵖ) : IsScalarTower k Bᵐᵒᵖ N where
smul_assoc r b n := by
change (r • b) • n = (algebraMap k Bᵐᵒᵖ r) • (b • n)
rw [Algebra.smul_def, mul_smul]
attribute [local instance] ExternalTensor.bimodule ExternalTensor.rightModule
abbrev Enveloping := B ⊗[k] Bᵐᵒᵖ
abbrev RightComplex := CochainComplex (ModuleCat.{u} Bᵐᵒᵖ) ℤ
abbrev BiComplex := CochainComplex (ModuleCat.{u} (Enveloping k B)) ℤ
/-- Induction by the actual left ideal Be, on right modules and maps. -/
def columnFunctor (e : B) : ModuleCat.{u} Bᵐᵒᵖ ⥤ ModuleCat.{u} (Enveloping k B) where
obj N := ModuleCat.of _ (Column k B N e)
map f := ModuleCat.ofHom (columnMap k e f.hom)
map_id N := by apply ModuleCat.hom_ext; exact map_id
map_comp f g := by
apply ModuleCat.hom_ext
simpa [columnMap] using (map_comp (k := k) (LinearMap.id) (LinearMap.id) f.hom g.hom)
instance columnFunctor_additive (e : B) : (columnFunctor k B e).Additive where
map_add := by intros; apply ModuleCat.hom_ext; exact map_add_right _ _ _
instance columnFunctor_linear (e : B) : (columnFunctor k B e).Linear k where
map_smul f r := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
change ExternalTensor.map (k := k) LinearMap.id (r • f.hom) x =
(algebraMap k (Enveloping k B) r) • (columnMap k e f.hom x)
rw [map_smul_right]
exact (IsScalarTower.algebraMap_smul (Enveloping k B) r _).symm
abbrev columnComplex (e : B) : RightComplex B ⥤ BiComplex k B :=
(columnFunctor k B e).mapHomologicalComplex (ComplexShape.up ℤ)
/-- Right multiplication on the left tensor factor, natural in the right module. -/
def pathNat (ei ej a : B) (ha : a * ei = a) :
columnFunctor k B ej ⟶ columnFunctor k B ei where
app N := ModuleCat.ofHom (columnPath k ei ej a ha)
naturality {N N'} f := by
apply ModuleCat.hom_ext
ext x
induction x using TensorProduct.inductionOn with
| tmul c n => rfl
| add x y hx hy =>
exact (map_add _ _ _).trans ((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm)
abbrev path (ei ej a : B) (ha : a * ei = a) (D : RightComplex B) :
(columnComplex k B ej).obj D ⟶ (columnComplex k B ei).obj D :=
((pathNat k B ei ej a ha).mapHomologicalComplex (ComplexShape.up ℤ)).app D
@[reassoc] theorem path_naturality (ei ej a : B) (ha : a * ei = a)
{D E : RightComplex B} (f : D ⟶ E) :
(columnComplex k B ej).map f ≫ path k B ei ej a ha E =
path k B ei ej a ha D ≫ (columnComplex k B ei).map f :=
((pathNat k B ei ej a ha).mapHomologicalComplex (ComplexShape.up ℤ)).naturality f
@[simp] theorem sum_f {C D : BiComplex k B} {I : Type} (s : Finset I)
(f : I → (C ⟶ D)) (n : ℤ) : (∑ i ∈ s, f i).f n = ∑ i ∈ s, (f i).f n :=
map_sum (Hom.fAddMonoidHom n) f s
@[simp] theorem sum_hom {S : Type*} [Ring S] {X Y : ModuleCat S}
{I : Type} (s : Finset I) (f : I → (X ⟶ Y)) :
(∑ i ∈ s, f i).hom = ∑ i ∈ s, (f i).hom :=
map_sum ModuleCat.homAddEquiv f s
private theorem sum_smul_component_apply {R : Type u} [Ring R] [Algebra k R]
{K L : CochainComplex (ModuleCat.{u} R) ℤ} {I J : Type} [Fintype I] [Fintype J]
(r : I → J → k) (f : I → J → (K ⟶ L)) (n : ℤ) (x : K.X n) :
((∑ i, ∑ j, r i j • f i j).f n).hom x =
∑ i, ∑ j, algebraMap k R (r i j) • ((f i j).f n).hom x := by
have eval_sum {I : Type} (s : Finset I) (g : I → (K ⟶ L)) :
(∑ i ∈ s, g i).f n = ∑ i ∈ s, (g i).f n :=
map_sum (Hom.fAddMonoidHom n) g s
simp only [eval_sum, smul_f_apply, ModuleCat.hom_sum,
ModuleCat.hom_smul, LinearMap.sum_apply, LinearMap.smul_apply]
rfl
/-- The vertex-zero component vanishes by the actual quadratic relation in B. -/
theorem paths_relation {A J : Type} [Fintype A] [Fintype J]
(e₀ e₁ e₂ : B) (a : A → B) (b : J → B)
(ha : ∀ i, a i * e₀ = a i) (hb : ∀ j, b j * e₁ = b j)
(r : A → J → k) (hr : ∑ i, ∑ j, r i j • (b j * a i) = 0)
(D : RightComplex B) :
∑ i, ∑ j, r i j • (path k B e₁ e₂ (b j) (hb j) D ≫
path k B e₀ e₁ (a i) (ha i) D) = 0 := by
apply Hom.ext
funext n
apply ModuleCat.hom_ext
apply LinearMap.ext
intro x
induction x using TensorProduct.inductionOn with
| add x y hx hy =>
exact (map_add _ _ _).trans ((congrArg₂ (· + ·) hx hy).trans (map_add _ _ _).symm)
| tmul c v =>
erw [sum_smul_component_apply]
change (∑ i, ∑ j, (algebraMap k (Enveloping k B) (r i j)) •
((pathMap e₀ e₁ (a i) (ha i) (pathMap e₁ e₂ (b j) (hb j) c)) ⊗ₜ[k] v)) = 0
simp only [IsScalarTower.algebraMap_smul, TensorProduct.smul_tmul',
← TensorProduct.sum_tmul]
have hz : (∑ i, ∑ j, r i j •
pathMap e₀ e₁ (a i) (ha i) (pathMap e₁ e₂ (b j) (hb j) c)) = 0 := by
apply Subtype.ext
change ((leftIdeal B e₀).subtype.restrictScalars k)
(∑ i, ∑ j, r i j • pathMap e₀ e₁ (a i) (ha i)
(pathMap e₁ e₂ (b j) (hb j) c)) = 0
simp only [map_sum, map_smul, LinearMap.restrictScalars_apply,
Submodule.subtype_apply, pathMap_apply]
calc
_ = (c : B) * (∑ i, ∑ j, r i j • (b j * a i)) := by
simp only [Finset.mul_sum, Algebra.mul_smul_comm, mul_assoc]
_ = 0 := by rw [hr, mul_zero]
rw [hz, TensorProduct.zero_tmul]
end ExternalTensor.Section4
end
end OAI
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