| import OAI.Algebra.Finitistic.Selected.SelectedPDBarConsumer |
| import OAI.Algebra.Finitistic.Selected.SelectedPDResidueProjection |
| import OAI.Algebra.Finitistic.Selected.SelectedCountCohomology |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.SelectedPDDetection |
| open SelectedOrdinaryAlgebra SelectedOrdinaryModules |
| local instance : HasDerivedCategory.{1} (ModuleCat.{0} D) := |
| HasDerivedCategory.standard (ModuleCat.{0} D) |
|
|
| /-- A supplied inclusion/retraction of the original count summand into the |
| actual full bar converts native iterate survival directly into the pd bound. |
| Native supplies the proved nonzero degree; no shift/amplitude proof is duplicated. -/ |
| theorem lower_bound_of_surviving_count (Y : FiniteSelectionFamily.S.{0}) (r : ℕ) |
| (h : ¬ IsZero (SelectedBaseIterates.object Y r)) |
| (i : SelectedCountCohomology.count Y r ⟶ PDBarQuotient.baseDerived augmentation (Inflated Y)) |
| (p : PDBarQuotient.baseDerived augmentation (Inflated Y) ⟶ SelectedCountCohomology.count Y r) |
| (hip : i ≫ p = 𝟙 (SelectedCountCohomology.count Y r)) : |
| r ≤ projectiveDimension (module Y) := by |
| obtain ⟨n,hrn,_,hn⟩ := SelectedCountCohomology.nonzero_degree Y r h |
| exact lower_bound_of_base_derived_bar Y _ _ (Iso.refl _) i p hip r n hrn hn |
| end LittleFinitistic.SelectedPDDetection |
|
|
| end |
|
|
| end OAI |
|
|
