| import OAI.Algebra.Finitistic.Selected.SelectedRectification |
| import OAI.Algebra.Finitistic.Localization.TensorCohomologyDiagram |
| import OAI.Algebra.Finitistic.Rectification.FullTensorProperties |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| namespace LittleFinitistic.SelectedTensor |
| open SelectedFixedDiagram SelectedRectification QuadraticEncoding |
| variable (N : Type) [AddCommGroup N] [Module ℂ N] [Module (B P) N] |
|
|
| / |
| abbrev complex := FullTensorRectification.tensor P D0 D1 D2 f g relations N |
| abbrev source (v : Fin 3) := FullTensorRectification.source P D0 D1 D2 N v |
| abbrev comparison (v : Fin 3) (n : ℤ) := |
| TensorCohomologyDiagram.comparison P D0 D1 D2 f g relations N v n |
|
|
| / |
| instance comparison_isIso (v : Fin 3) (n : ℤ) : IsIso (comparison N v n) := by |
| obtain ⟨l₀,h₀,b₀⟩ := UnderlyingRepresentatives.bounded P fixed.D0 |
| obtain ⟨l₁,h₁,b₁⟩ := UnderlyingRepresentatives.bounded P fixed.D1 |
| obtain ⟨l₂,h₂,b₂⟩ := UnderlyingRepresentatives.bounded P fixed.D2 |
| let lo := min l₀ (min l₁ l₂) |
| let hi := max h₀ (max h₁ h₂) |
| refine TensorCohomologyDiagram.comparison_isIso P D0 D1 D2 f g relations N v n lo hi |
| (fun n hn => b₀ n ?_) (fun n hn => b₁ n ?_) (fun n hn => b₂ n ?_) |
| (UnderlyingRepresentatives.projective P fixed.D0) |
| (UnderlyingRepresentatives.projective P fixed.D1) |
| (UnderlyingRepresentatives.projective P fixed.D2) |
| all_goals dsimp [lo,hi] at hn; omega |
| |
| def comparisonIso (v : Fin 3) (n : ℤ) := asIso (comparison N v n) |
| |
| / |
| theorem bounded : ∃ lo hi : ℤ, ∀ n, n < lo ∨ hi < n → IsZero ((complex N).X n) := by |
| obtain ⟨lo,hi,h⟩ := SelectedRectification.verified |
| exact ⟨lo-2,hi,FullTensorProperties.bounded ℂ (B P) N bimodule (lo-2) hi h.bounded⟩ |
| |
| / |
| theorem finite [FiniteDimensional ℂ N] (n : ℤ) : |
| FiniteDimensional ℂ ((ModuleCat.restrictScalars (algebraMap ℂ (B P))).obj ((complex N).X n)) := by |
| obtain ⟨lo,hi,h⟩ := SelectedRectification.verified |
| have : FiniteDimensional ℂ ((ExternalTensor.Section4.forgetK ℂ (B P)).obj (bimodule.X n)) := h.finite n |
| exact FullTensorProperties.finite ℂ (B P) N (bimodule.X n) |
| end LittleFinitistic.SelectedTensor |
| |
| end |
| |
| end OAI |
| |
