| import Mathlib.Algebra.Category.ModuleCat.Biproducts |
| import OAI.Algebra.Finitistic.Rectification.InducedDiagram |
| import OAI.Algebra.Finitistic.Rectification.ColumnProperties |
| import OAI.Algebra.Finitistic.Rectification.RectificationTotal |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.InducedRectification |
| open ExternalTensor.Section4 |
| universe u |
| variable {k B : Type u} [Field k] [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) |
|
|
| /-- The actual three-column induced bimodule complex of Section 4. -/ |
| def total : BiComplex k B := |
| RectificationTotal.total (p s D₀ D₁ D₂ f g) (q s D₀ D₁ D₂ f g) |
| (nullHomotopy s D₀ D₁ D₂ f g H) |
|
|
| /-- Its right-module restriction, retaining the original bimodule construction. -/ |
| def rightTotal : RightComplex B := |
| ((forgetRight k B).mapHomologicalComplex (.up ℤ)).obj (total s D₀ D₁ D₂ f g H) |
|
|
| theorem total_bounded (lo hi : ℤ) |
| (h₀ : ∀ n, n < lo ∨ hi < n → IsZero (D₀.X n)) |
| (h₁ : ∀ n, n < lo ∨ hi < n → IsZero (D₁.X n)) |
| (h₂ : ∀ n, n < lo ∨ hi < n → IsZero (D₂.X n)) : |
| ∀ n, n < lo-2 ∨ hi < n → IsZero ((total s D₀ D₁ D₂ f g H).X n) := by |
| have hc := columns_bounded s D₀ D₁ D₂ f g lo hi h₀ h₁ h₂ |
| exact RectificationTotal.total_bounded _ _ _ lo hi |
| (fun n hn => (hc n hn).1) (fun n hn => (hc n hn).2.1) |
| (fun n hn => (hc n hn).2.2) |
|
|
| /-- Termwise projectivity holds for the actual right restriction of the total. -/ |
| theorem rightTotal_projective |
| (h₀ : ∀ n, Module.Projective Bᵐᵒᵖ (D₀.X n)) |
| (h₁ : ∀ n, Module.Projective Bᵐᵒᵖ (D₁.X n)) |
| (h₂ : ∀ n, Module.Projective Bᵐᵒᵖ (D₂.X n)) : |
| ∀ n, Projective ((rightTotal s D₀ D₁ D₂ f g H).X n) := by |
| have hc := columns_right_projective s D₀ D₁ D₂ f g h₀ h₁ h₂ |
| exact RectificationTotal.total_projective_after _ _ _ (forgetRight k B) |
| (fun n => (hc n).1) (fun n => (hc n).2.1) (fun n => (hc n).2.2) |
|
|
| /-- The entire zero column has its original chain inclusion. -/ |
| def inclusion : L0Prime s D₀ D₁ D₂ f g ⟶ total s D₀ D₁ D₂ f g H := |
| RectificationTotal.inclusion _ _ _ |
|
|
| /-- The prescribed first boundary becomes null-homotopic in the actual total. -/ |
| def boundaryHomotopy : Homotopy |
| (p s D₀ D₁ D₂ f g ≫ inclusion s D₀ D₁ D₂ f g H) 0 := |
| RectificationTotal.boundaryHomotopy _ _ _ |
| end LittleFinitistic.InducedRectification |
|
|
| end |
|
|
| end OAI |
|
|
