| import OAI.Algebra.Finitistic.Rectification.SourceFiltrationCone |
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| namespace OAI |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex CochainComplex |
| open LittleFinitistic.RectificationTotal |
| open scoped TensorProduct ModuleCat.Algebra |
| noncomputable section |
| namespace ExternalTensor.Section4.SourceFiltration.Exactness |
| universe u |
| variable {k B : Type u} [Field k] [Ring B] [Algebra k B] |
| variable {A J R : Type} [Fintype A] [Fintype J] [Fintype R] |
| variable (s : Quadratic k B (A := A) (J := J) (R := R)) |
| variable (D₀ D₁ D₂ : RightComplex B) |
| variable (f : A → (D₀ ⟶ D₁)) (g : J → (D₁ ⟶ D₂)) |
| variable (H : ∀ r, Homotopy (relationMap s D₀ D₁ D₂ f g r) 0) |
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| abbrev data := inducedData s D₀ D₁ D₂ f g |
| abbrev P := total (p s D₀ D₁ D₂ f g) (q s D₀ D₁ D₂ f g) |
| (nullHomotopy s D₀ D₁ D₂ f g H) |
| abbrev F1 := total (p1 (data s D₀ D₁ D₂ f g)) 0 (h1 (data s D₀ D₁ D₂ f g)) |
| abbrev G0 := total (p0 (data s D₀ D₁ D₂ f g)) (q0 (data s D₀ D₁ D₂ f g)) |
| (h0 (data s D₀ D₁ D₂ f g)) |
| abbrev G1 := total (pGrade1 (data s D₀ D₁ D₂ f g)) 0 |
| (hGrade1 (data s D₀ D₁ D₂ f g)) |
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| abbrev i1 : F1 s D₀ D₁ D₂ f g ⟶ P s D₀ D₁ D₂ f g H := |
| (inducedInclude1 s D₀ D₁ D₂ f g H).totalMap |
| abbrev pi0 : P s D₀ D₁ D₂ f g H ⟶ G0 s D₀ D₁ D₂ f g := |
| (inducedQuotient0 s D₀ D₁ D₂ f g H).totalMap |
| abbrev i2 : (columnComplex k B s.e₂).obj D₂ ⟶ F1 s D₀ D₁ D₂ f g := |
| v2Inclusion (data s D₀ D₁ D₂ f g) |
| abbrev pi1 : F1 s D₀ D₁ D₂ f g ⟶ G1 s D₀ D₁ D₂ f g := |
| (quotient1 (data s D₀ D₁ D₂ f g)).totalMap |
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|
| /-- Section 4's actual two nontrivial source-filtration steps. No decomposition, |
| exactness, extra coherence, vertex equivalence, or main theorem is an input. -/ |
| theorem exact_source_filtration (n : ℤ) : |
| (Function.Injective ((i1 s D₀ D₁ D₂ f g H).f n) ∧ |
| Function.Exact ((i1 s D₀ D₁ D₂ f g H).f n) ((pi0 s D₀ D₁ D₂ f g H).f n) ∧ |
| Function.Surjective ((pi0 s D₀ D₁ D₂ f g H).f n)) ∧ |
| (Function.Injective ((i2 s D₀ D₁ D₂ f g).f n) ∧ |
| Function.Exact ((i2 s D₀ D₁ D₂ f g).f n) ((pi1 s D₀ D₁ D₂ f g).f n) ∧ |
| Function.Surjective ((pi1 s D₀ D₁ D₂ f g).f n)) ∧ |
| i2 s D₀ D₁ D₂ f g ≫ i1 s D₀ D₁ D₂ f g H = |
| (data s D₀ D₁ D₂ f g).in₂ ≫ |
| inclusion (p s D₀ D₁ D₂ f g) (q s D₀ D₁ D₂ f g) |
| (nullHomotopy s D₀ D₁ D₂ f g H) := |
| ⟨induced_source0_shortExact s D₀ D₁ D₂ f g H n, |
| v2_shortExact (data s D₀ D₁ D₂ f g) n, |
| v2_nested (data s D₀ D₁ D₂ f g) (nullHomotopy s D₀ D₁ D₂ f g H)⟩ |
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| theorem categorical_source0 (n : ℤ) : |
| let E := (exact_source_filtration s D₀ D₁ D₂ f g H n).1 |
| (ModuleCat.shortComplexOfCompEqZero ((i1 s D₀ D₁ D₂ f g H).f n).hom |
| ((pi0 s D₀ D₁ D₂ f g H).f n).hom E.2.1.linearMap_comp_eq_zero).ShortExact := by |
| let E := (exact_source_filtration s D₀ D₁ D₂ f g H n).1 |
| exact ModuleCat.shortComplex_shortExact _ E.2.1 E.1 E.2.2 |
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| omit [Fintype R] in |
| theorem categorical_source1 (n : ℤ) : |
| let E := v2_shortExact (data s D₀ D₁ D₂ f g) n |
| (ModuleCat.shortComplexOfCompEqZero ((i2 s D₀ D₁ D₂ f g).f n).hom |
| ((pi1 s D₀ D₁ D₂ f g).f n).hom E.2.1.linearMap_comp_eq_zero).ShortExact := by |
| let E := v2_shortExact (data s D₀ D₁ D₂ f g) n |
| exact ModuleCat.shortComplex_shortExact _ E.2.1 E.1 E.2.2 |
| end ExternalTensor.Section4.SourceFiltration.Exactness |
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| end |
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| end OAI |
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