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ae00908 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | 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
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