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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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | 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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