| import OAI.Algebra.Finitistic.Rectification.FiltrationExactness |
| import OAI.Algebra.Finitistic.Bimodule.BimoduleVertex |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| namespace LittleFinitistic.ModuleComplexExact |
| universe u |
| variable {S : Type u} [Ring S] |
| variable {X Y Z : CochainComplex (ModuleCat.{u} S) ℤ} (f : X ⟶ Y) (g : Y ⟶ Z) |
|
|
| theorem comp_zero (h : ∀ n, Function.Exact (f.f n) (g.f n)) : f ≫ g = 0 := by |
| ext n x |
| exact (h n).apply_apply_eq_zero x |
|
|
| def sequence (h : ∀ n, Function.Exact (f.f n) (g.f n)) : |
| ShortComplex (CochainComplex (ModuleCat.{u} S) ℤ) := .mk f g (comp_zero f g h) |
|
|
| theorem shortExact (h : ∀ n, Function.Injective (f.f n) ∧ |
| Function.Exact (f.f n) (g.f n) ∧ Function.Surjective (g.f n)) : |
| (sequence f g (fun n => (h n).2.1)).ShortExact := by |
| apply HomologicalComplex.shortExact_of_degreewise_shortExact |
| intro n |
| exact ModuleCat.shortComplex_shortExact _ (h n).2.1 (h n).1 (h n).2.2 |
| end LittleFinitistic.ModuleComplexExact |
|
|
| namespace ExternalTensor.Section4.SourceFiltration.Complex |
| 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) |
| open Exactness LittleFinitistic |
|
|
| def source0 := ModuleComplexExact.sequence (i1 s D₀ D₁ D₂ f g H) |
| (pi0 s D₀ D₁ D₂ f g H) |
| (fun n => (exact_source_filtration s D₀ D₁ D₂ f g H n).1.2.1) |
|
|
| def source1 := ModuleComplexExact.sequence (i2 s D₀ D₁ D₂ f g) |
| (pi1 s D₀ D₁ D₂ f g) |
| (fun n => (v2_shortExact (data s D₀ D₁ D₂ f g) n).2.1) |
|
|
| theorem source0_shortExact : (source0 s D₀ D₁ D₂ f g H).ShortExact := |
| ModuleComplexExact.shortExact _ _ |
| (fun n => (exact_source_filtration s D₀ D₁ D₂ f g H n).1) |
|
|
| omit [Fintype R] in |
| theorem source1_shortExact : (source1 s D₀ D₁ D₂ f g).ShortExact := |
| ModuleComplexExact.shortExact _ _ (v2_shortExact (data s D₀ D₁ D₂ f g)) |
|
|
| /-- Every original source filtration remains short exact after taking a left vertex. -/ |
| theorem vertex_shortExact (e : B) (he : e*e=e) |
| (S : ShortComplex (BiComplex k B)) (hS : S.ShortExact) : |
| (S.map ((BimoduleVertex.functor k B e).mapHomologicalComplex (.up ℤ))).ShortExact := by |
| apply HomologicalComplex.shortExact_of_degreewise_shortExact |
| intro n |
| exact BimoduleVertex.map_shortExact k B e he |
| (S.map (HomologicalComplex.eval _ _ n)) |
| ((HomologicalComplex.shortExact_iff_degreewise_shortExact S).1 hS n) |
| end ExternalTensor.Section4.SourceFiltration.Complex |
|
|
| end |
|
|
| end OAI |
|
|
