| import OAI.Algebra.Finitistic.Rectification.GradedVertexAcyclic |
| import OAI.Algebra.Finitistic.Rectification.FirstVertex |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| namespace LittleFinitistic.SecondVertex |
| 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 "T" => QuadraticEncoding.B P |
| local notation "s" => ConcreteQuadraticDiagram.quadratic P |
| abbrev vertex := BimoduleVertex.functor k (QuadraticEncoding.B P) |
| (ThreeVertexAlgebra.vertex (QuadraticEncoding.composition P) 1) |
| abbrev vertexComplex := (vertex P).mapHomologicalComplex (.up ℤ) |
|
|
| def intoF1 : (columnComplex k T (s).e₁).obj D₁ ⟶ Exactness.F1 s D₀ D₁ D₂ f g := |
| biprod.inl ≫ RectificationTotal.inclusion (p1 (Exactness.data s D₀ D₁ D₂ f g)) 0 |
| (h1 (Exactness.data s D₀ D₁ D₂ f g)) |
|
|
| theorem intoF1_isIso : IsIso ((vertexComplex P).map (intoF1 P D₀ D₁ D₂ f g)) := by |
| let F := vertex P |
| let G := vertexComplex P |
| have hz : IsZero (G.obj (Z (k := k) (B := T))) := by |
| apply G.map_isZero |
| apply ZeroColumnTotal.complex_zero |
| intro n |
| exact ModuleCat.isZero_iff_subsingleton.mpr inferInstance |
| have hv : IsZero (G.obj ((columnComplex k T (s).e₂).obj D₂)) := |
| VertexColumnComplex.reverse_isZero (QuadraticEncoding.composition P) 1 2 (by decide) D₂ |
| have hs : IsZero (G.obj (F1₁ (Exactness.data s D₀ D₁ D₂ f g))) := by |
| apply (G.mapBiproduct _).isZero_iff.mpr |
| exact sum_isZero _ (fun _ => |
| VertexColumnComplex.reverse_isZero (QuadraticEncoding.composition P) 1 2 (by decide) D₁) |
| have hi := ZeroColumnTotal.mapped_inclusion_isIso |
| (p1 (Exactness.data s D₀ D₁ D₂ f g)) 0 (h1 (Exactness.data s D₀ D₁ D₂ f g)) F |
| (fun n => (HomologicalComplex.eval _ _ n).map_isZero hs) |
| (fun n => (HomologicalComplex.eval _ _ n).map_isZero hz) |
| have hj := FirstVertex.mapped_inl_isIso G |
| ((columnComplex k T (s).e₁).obj D₁) ((columnComplex k T (s).e₂).obj D₂) hv |
| let := hi |
| let := hj |
| dsimp [intoF1] |
| erw [Functor.map_comp] |
| exact (show IsIso (G.map (biprod.inl : |
| (columnComplex k T (s).e₁).obj D₁ ⟶ F1₀ (Exactness.data s D₀ D₁ D₂ f g)) ≫ |
| G.map (RectificationTotal.inclusion (p1 (Exactness.data s D₀ D₁ D₂ f g)) 0 |
| (h1 (Exactness.data s D₀ D₁ D₂ f g)))) from by |
| have : IsIso (G.map (biprod.inl : |
| (columnComplex k T (s).e₁).obj D₁ ⟶ F1₀ (Exactness.data s D₀ D₁ D₂ f g))) := hj |
| rw [isIso_comp_left_iff] |
| exact hi) |
|
|
| variable (H : ∀ r, Homotopy (relationMap (ConcreteQuadraticDiagram.quadratic P) D₀ D₁ D₂ f g r) 0) |
|
|
| 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).e₁) ((ThreeVertexAlgebra.directedSystem (QuadraticEncoding.composition P)).idem 1) S |
| (Complex.source0_shortExact s D₀ D₁ D₂ f g H)) |
| (GradedVertexAcyclic.grade0_vertex1_acyclic P D₀ D₁ D₂ f g) |
|
|
| def iota : D₁ ⟶ (vertexComplex P).obj (Exactness.P s D₀ D₁ D₂ f g H) := |
| (VertexColumnComplex.diagonalComplexIso (QuadraticEncoding.composition P) 1 D₁).hom ≫ |
| (vertexComplex P).map (intoF1 P 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 := intoF1_isIso P D₀ D₁ D₂ f g |
| let := filtration_inclusion_quasiIso P D₀ D₁ D₂ f g H |
| let e : D₁ ≅ (vertexComplex P).obj ((columnComplex k T (s).e₁).obj D₁) := |
| VertexColumnComplex.diagonalComplexIso (QuadraticEncoding.composition P) 1 D₁ |
| change QuasiIso (e.hom ≫ (vertexComplex P).map (intoF1 P D₀ D₁ D₂ f g) ≫ |
| (vertexComplex P).map (Exactness.i1 s D₀ D₁ D₂ f g H)) |
| infer_instance |
|
|
| /-- The filtration route retains the original vertex-one summand insertion. -/ |
| theorem intoF1_nested : intoF1 P D₀ D₁ D₂ f g ≫ Exactness.i1 s D₀ D₁ D₂ f g H = |
| (Exactness.data s D₀ D₁ D₂ f g).in₁ ≫ |
| InducedRectification.inclusion s D₀ D₁ D₂ f g H := by |
| exact (Category.assoc _ _ _).trans (congrArg |
| (fun h => (biprod.inl : (columnComplex k T (s).e₁).obj D₁ ⟶ |
| F1₀ (Exactness.data s D₀ D₁ D₂ f g)) ≫ h) |
| ((include1 (inducedData s D₀ D₁ D₂ f g) (nullHomotopy s D₀ D₁ D₂ f g H)).inclusion_totalMap)) |
|
|
| theorem iota_original : iota P D₀ D₁ D₂ f g H = |
| (VertexColumnComplex.diagonalComplexIso (QuadraticEncoding.composition P) 1 D₁).hom ≫ |
| (vertexComplex P).map ((Exactness.data s D₀ D₁ D₂ f g).in₁) ≫ |
| (vertexComplex P).map (InducedRectification.inclusion s D₀ D₁ D₂ f g H) := by |
| dsimp only [iota] |
| erw [← (vertexComplex P).map_comp (intoF1 P D₀ D₁ D₂ f g) |
| (Exactness.i1 s D₀ D₁ D₂ f g H), intoF1_nested, Functor.map_comp] |
| rfl |
| end LittleFinitistic.SecondVertex |
|
|
| end |
|
|
| end OAI |
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