| import OAI.Algebra.Finitistic.Rectification.ThirdVertexFiltration |
| import OAI.Algebra.Finitistic.Rectification.InducedRectification |
| import OAI.Algebra.Finitistic.Rectification.FinalGradedAcyclic |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| namespace LittleFinitistic.ThirdVertex |
| open ExternalTensor.Section4 ExternalTensor.Section4.SourceFiltration |
| universe u |
| variable {k : Type u} [Field k] {d t : ℕ} (P : QuadraticEncoding.Relations k d t) |
| variable (D₀ D₁ D₂ : RightComplex (QuadraticEncoding.B P)) |
| variable (f : Fin (d+1) → (D₀ ⟶ D₁)) (g : Fin (d+1) → (D₁ ⟶ D₂)) |
| variable (H : ∀ r, Homotopy (relationMap (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g r) 0) |
| local notation "s" => ConcreteQuadraticDiagram.quadratic P |
| abbrev vertexComplex := AdjacentArrowSum.vertexComplex (QuadraticEncoding.composition P) 2 |
|
|
| theorem filtration_inclusion_quasiIso : |
| QuasiIso ((vertexComplex P).map (Exactness.i1 s D₀ D₁ D₂ f g H)) := by |
| let S := Complex.source0 s D₀ D₁ D₂ f g H |
| exact RectificationAcyclic.quasiIso_of_acyclic_quotient (S.map (vertexComplex P)) |
| (Complex.vertex_shortExact ((s).e₂) |
| ((ThreeVertexAlgebra.directedSystem (QuadraticEncoding.composition P)).idem 2) S |
| (Complex.source0_shortExact s D₀ D₁ D₂ f g H)) |
| (FinalGradedAcyclic.grade0_vertex2_acyclic P D₀ D₁ D₂ f g) |
|
|
| def iota : D₂ ⟶ (vertexComplex P).obj (Exactness.P s D₀ D₁ D₂ f g H) := |
| (VertexColumnComplex.diagonalComplexIso (QuadraticEncoding.composition P) 2 D₂).hom ≫ |
| (vertexComplex P).map (Exactness.i2 s D₀ D₁ D₂ f g) ≫ |
| (vertexComplex P).map (Exactness.i1 s D₀ D₁ D₂ f g H) |
|
|
| theorem iota_quasiIso : QuasiIso (iota P D₀ D₁ D₂ f g H) := by |
| let := ThirdVertexFiltration.inclusion_quasiIso P D₀ D₁ D₂ f g |
| let := filtration_inclusion_quasiIso P D₀ D₁ D₂ f g H |
| let e : D₂ ≅ (vertexComplex P).obj ((columnComplex k (QuadraticEncoding.B P) (s).e₂).obj D₂) := |
| VertexColumnComplex.diagonalComplexIso (QuadraticEncoding.composition P) 2 D₂ |
| change QuasiIso (e.hom ≫ (vertexComplex P).map (Exactness.i2 s D₀ D₁ D₂ f g) ≫ |
| (vertexComplex P).map (Exactness.i1 s D₀ D₁ D₂ f g H)) |
| infer_instance |
|
|
| /-- The final-vertex map is the original e2 tensor insertion followed by the original total inclusion. -/ |
| theorem iota_original : iota P D₀ D₁ D₂ f g H = |
| (VertexColumnComplex.diagonalComplexIso (QuadraticEncoding.composition P) 2 D₂).hom ≫ |
| (vertexComplex P).map ((Exactness.data s D₀ D₁ D₂ f g).in₂) ≫ |
| (vertexComplex P).map (InducedRectification.inclusion s D₀ D₁ D₂ f g H) := by |
| dsimp only [iota] |
| erw [← (vertexComplex P).map_comp (Exactness.i2 s D₀ D₁ D₂ f g) |
| (Exactness.i1 s D₀ D₁ D₂ f g H), |
| (Exactness.exact_source_filtration s D₀ D₁ D₂ f g H 0).2.2, Functor.map_comp] |
| rfl |
| end LittleFinitistic.ThirdVertex |
|
|
| end |
|
|
| end OAI |
|
|
