| import OAI.Algebra.Finitistic.Rectification.RectificationTotal |
| import OAI.Algebra.Finitistic.Rectification.ColumnFinite |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex CochainComplex |
| noncomputable section |
| namespace ExternalTensor.Section4 |
| universe u |
| variable {k : Type u} [Field k] |
| section General |
| variable {C : Type*} [Category C] [Preadditive C] [HasBinaryBiproducts C] |
| variable {L₀ L₁ L₂ : CochainComplex C ℤ} (p : L₁ ⟶ L₀) (q : L₂ ⟶ L₁) |
| (h : Homotopy (q ≫ p) 0) |
|
|
| / |
| an additive scalar restriction makes all three columns finite-dimensional. -/ |
| theorem total_finite_after (F : C ⥤ ModuleCat.{u} k) [F.Additive] |
| (h₀ : ∀ i, FiniteDimensional k (F.obj (L₀.X i))) |
| (h₁ : ∀ i, FiniteDimensional k (F.obj (L₁.X i))) |
| (h₂ : ∀ i, FiniteDimensional k (F.obj (L₂.X i))) (i : ℤ) : |
| FiniteDimensional k (F.obj ((LittleFinitistic.RectificationTotal.total p q h).X i)) := by |
| let := preservesBinaryBiproducts_of_preservesBiproducts F |
| apply finite_of_iso k (F.mapIso (LittleFinitistic.RectificationTotal.termIso p q h i) ≪≫ |
| F.mapBiprod _ _ ≪≫ biprod.mapIso (F.mapBiprod _ _) (Iso.refl _)) |
| exact biprod_finite k (biprod_finite k (h₂ (i+2)) (h₁ (i+1))) (h₀ i) |
|
|
| end General |
|
|
| variable {B : Type u} [Ring B] [Algebra 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)) |
| variable (H : ∀ r, Homotopy (relationMap s D₀ D₁ D₂ f g r) 0) |
|
|
| / |
| with no projectivity or finite-dimensionality assumption on P or its columns. -/ |
| theorem induced_total_finite [FiniteDimensional k B] |
| (h₀ : ∀ i, Module.Finite Bᵐᵒᵖ (D₀.X i)) |
| (h₁ : ∀ i, Module.Finite Bᵐᵒᵖ (D₁.X i)) |
| (h₂ : ∀ i, Module.Finite Bᵐᵒᵖ (D₂.X i)) (i : ℤ) : |
| FiniteDimensional k ((forgetK k B).obj |
| ((LittleFinitistic.RectificationTotal.total (p s D₀ D₁ D₂ f g) |
| (q s D₀ D₁ D₂ f g) (nullHomotopy s D₀ D₁ D₂ f g H)).X i)) := by |
| exact total_finite_after _ _ _ (forgetK k B) |
| (fun n => (columns_finite s D₀ D₁ D₂ f g h₀ h₁ h₂ n).1) |
| (fun n => (columns_finite s D₀ D₁ D₂ f g h₀ h₁ h₂ n).2.1) |
| (fun n => (columns_finite s D₀ D₁ D₂ f g h₀ h₁ h₂ n).2.2) i |
| end ExternalTensor.Section4 |
|
|
| end |
|
|
| end OAI |
|
|
