| import OAI.Algebra.Finitistic.Selected.SelectedCohomologyVanishing | |
| import OAI.Algebra.Finitistic.Localization.TwoDegreeFormalityNaturality | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Limits | |
| attribute [local instance] HasDerivedCategory.standard HasExt.standard | |
| namespace LittleFinitistic.SelectedTensorSplitting | |
| open SelectedFixedDiagram QuadraticEncoding | |
| variable (N : Type) [AddCommGroup N] [Module ℂ N] [Module (B P) N] | |
| /-- The actual derived object represented by the full B-valued fixed tensor complex. -/ | |
| abbrev object : DerivedCategory (ModuleCat (B P)) := DerivedCategory.Q.obj (SelectedTensor.complex N) | |
| def homologyIso (n : ℤ) : | |
| (DerivedCategory.homologyFunctor (ModuleCat (B P)) n).obj (object N) ≅ | |
| (SelectedTensor.complex N).homology n := | |
| (DerivedCategory.homologyFunctorFactors (ModuleCat (B P)) n).app (SelectedTensor.complex N) | |
| /-- Support of the evaluated representative diagram implies actual derived | |
| cohomological support of the full tensor object. -/ | |
| theorem twoDegrees | |
| (h : ∀ n : ℤ, n ≠ 0 → n ≠ -3 → ∀ v : Fin 3, | |
| IsZero ((SelectedTensor.source N v).homology n)) : | |
| DerivedTruncation.TwoDegrees (object N) := by | |
| intro n hn0 hn3 | |
| exact (homologyIso N n).isZero_iff.mpr | |
| (SelectedCohomologyVanishing.cohomology_isZero N n (h n hn0 hn3)) | |
| /-- Apply the proved B-linear truncation splitting to the actual fixed tensor, | |
| using full module cohomology models and source-diagram support. -/ | |
| def split (R : ModuleCat (B P)) | |
| (h0 : (SelectedTensor.complex N).homology 0 ≅ R) | |
| (h3 : (SelectedTensor.complex N).homology (-3) ≅ R) | |
| (hz : ∀ n : ℤ, n ≠ 0 → n ≠ -3 → ∀ v : Fin 3, | |
| IsZero ((SelectedTensor.source N v).homology n)) : | |
| object N ≅ (DerivedCategory.singleFunctor (ModuleCat (B P)) 0).obj R ⊞ | |
| ((DerivedCategory.singleFunctor (ModuleCat (B P)) 0).obj R)⟦(3 : ℤ)⟧ := | |
| TwoDegreeFormality.splitIsoWithModels P (object N) (twoDegrees N hz) R R | |
| (homologyIso N 0 ≪≫ h0) (homologyIso N (-3) ≪≫ h3) | |
| end LittleFinitistic.SelectedTensorSplitting | |
| end | |
| end OAI | |