| import OAI.Algebra.Finitistic.Rectification.GradedVertexAcyclic | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex | |
| namespace LittleFinitistic.ThirdVertexFiltration | |
| 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)) | |
| local notation "s" => ConcreteQuadraticDiagram.quadratic P | |
| abbrev vertexComplex := AdjacentArrowSum.vertexComplex (QuadraticEncoding.composition P) 2 | |
| /-- The original deepest source-filtration inclusion is a quasi-isomorphism at vertex two. -/ | |
| theorem inclusion_quasiIso : | |
| QuasiIso ((vertexComplex P).map (Exactness.i2 s D₀ D₁ D₂ f g)) := by | |
| let S := Complex.source1 s D₀ D₁ D₂ f g | |
| exact RectificationAcyclic.quasiIso_of_acyclic_quotient (S.map (vertexComplex P)) | |
| (Complex.vertex_shortExact ((s).e₂) | |
| ((ThreeVertexAlgebra.directedSystem (QuadraticEncoding.composition P)).idem 2) S | |
| (Complex.source1_shortExact s D₀ D₁ D₂ f g)) | |
| (GradedVertexAcyclic.grade1_vertex2_acyclic P D₀ D₁ D₂ f g) | |
| end LittleFinitistic.ThirdVertexFiltration | |
| end | |
| end OAI | |