| import OAI.Algebra.Finitistic.Selected.SelectedPDUpperConsumer |
| import OAI.Algebra.Finitistic.Bar.CountTorBounds |
| import OAI.Algebra.Finitistic.Selected.SelectedCountExtinction |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.SelectedPDConclusion |
| open SelectedOrdinaryAlgebra SelectedOrdinaryModules |
| local instance : HasDerivedCategory.{1} (ModuleCat.{0} D) := |
| HasDerivedCategory.standard (ModuleCat.{0} D) |
|
|
| /-- The canonical base-change Tor is computed by the actual selected bar, |
| independently of which compatible field-scalar instance presents that bar. -/ |
| def baseTorIso (Y : FiniteSelectionFamily.S.{0}) (n : ℕ) : |
| ((DerivedBar.Balanced.tensorFunctor augmentation).leftDerived n).obj (module Y) ≅ |
| (SelectedPDDetection.baseBar Y).homology n := |
| DerivedBar.Balanced.tensorBarHomologyIso augmentation (Inflated Y) n ≪≫ |
| (HomologicalComplex.homologyFunctor (ModuleCat D) (.down ℕ) n).mapIso |
| (DerivedBar.tensorBarIso ℂ A D (Inflated Y) augmentation) |
|
|
| /-- Actual tensor extinction implies finite projective dimension over the |
| one fixed square-zero algebra A. The bar bounds and minimal detector are proved. -/ |
| theorem finite_pd_of_extinction (Y : FiniteSelectionFamily.S.{0}) (t : ℕ) |
| (h : IsZero (SelectedBaseIterates.object Y t)) : |
| projectiveDimension (module Y) < ⊤ := by |
| apply SelectedPDDetection.finite_pd_of_bounded_base_bar Y |
| refine ⟨t * (3 * length + 5), fun n hn => ?_⟩ |
| have hc := DerivedBar.ParityBridge.baseChangeTor_bounded_of_counts ℂ D X |
| (SelectedBaseIterates.input Y) (t * (3 * length + 5)) |
| (fun r n hn => SelectedCountExtinction.uniform_homology_zero Y t h r (-(n : ℤ)) |
| (Or.inl (by omega))) n hn |
| exact hc.of_iso (baseTorIso Y n).symm |
|
|
| /-- A surviving actual derived iterate supplies a nonzero original count |
| class and hence the manuscript's projective-dimension lower bound. -/ |
| theorem lower_bound_of_survival (Y : FiniteSelectionFamily.S.{0}) (r : ℕ) |
| (h : ¬ IsZero (SelectedBaseIterates.object Y r)) : |
| (r : WithBot ℕ∞) ≤ projectiveDimension (module Y) := by |
| obtain ⟨n,hrn,_,hn⟩ := SelectedCountCohomology.nonzero_degree Y r h |
| have hc := DerivedBar.ParityBridge.baseChangeTor_nonzero_of_count ℂ D X |
| (SelectedBaseIterates.input Y) r n hn |
| apply SelectedPDDetection.lower_bound_of_base_bar Y r n hrn |
| intro hz |
| exact hc (hz.of_iso (baseTorIso Y n)) |
|
|
| /-- Both conclusions of the manuscript's projective-dimension detector for |
| the actual finite selected family; no bar comparison or pd bound is a premise. -/ |
| theorem detection (Y : FiniteSelectionFamily.S.{0}) : |
| (∀ t, IsZero (SelectedBaseIterates.object Y t) → projectiveDimension (module Y) < ⊤) ∧ |
| (∀ r, ¬ IsZero (SelectedBaseIterates.object Y r) → |
| (r : WithBot ℕ∞) ≤ projectiveDimension (module Y)) := |
| ⟨finite_pd_of_extinction Y, lower_bound_of_survival Y⟩ |
| end LittleFinitistic.SelectedPDConclusion |
| |
| end |
| |
| end OAI |
| |
