| import OAI.Algebra.Finitistic.Selection.FiniteSelectionFamily |
| import OAI.Algebra.Finitistic.Iteration.ObjectwiseSplitting |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| attribute [local instance] HasDerivedCategory.standard HasExt.standard |
| namespace LittleFinitistic.SelectedIteration |
| open SelectedFixedDiagram QuadraticEncoding FiniteSelectionFamily |
|
|
| variable (Phi : DerivedCategory (ModuleCat (B P)) ⥤ DerivedCategory (ModuleCat (B P))) |
| [Phi.Additive] [Phi.CommShift ℤ] |
| /- This is solely the remaining derived-tensor/stalk comparison. The realization |
| and exact reference cohomology have already been proved independently. -/ |
| variable (comparison : ∀ Y : S, Phi.obj (derived Y) ≅ |
| SelectedTensorSplitting.object (representation Y).Evaluation) |
| include comparison |
|
|
| def split (Y : S) : Phi.obj (derived Y) ≅ derived (H Y) ⊞ (derived (H Y))⟦(3 : ℤ)⟧ := |
| comparison Y ≪≫ FiniteSelectionFamily.realization Y |
|
|
| / |
| def iteration (j : ℕ) (Y : S) : |
| (Phi.obj^[j]) (derived Y) ≅ |
| ⨁ fun q : Fin (j+1) => ⨁ fun _ : Fin (Nat.choose j q.val) => |
| (derived ((H^[j]) Y))⟦(3 : ℤ)*(q.val : ℤ)⟧ := |
| ExternalTensor.Binomial.section4Iteration Phi derived H (split Phi comparison) j Y |
|
|
| private def shiftedSplit (Y : S) : |
| Phi.obj (derived Y) ≅ (derived (H Y))⟦(0 : ℤ)⟧ ⊞ (derived (H Y))⟦(0+3 : ℤ)⟧ := |
| split Phi comparison Y ≪≫ biprod.mapIso (shiftZero ℤ (derived (H Y))).symm (Iso.refl _) |
|
|
| theorem isZero_iff (j : ℕ) (Y : S) : |
| IsZero ((Phi.obj^[j]) (derived Y)) ↔ Subsingleton (selectionIterate Y.val j) := by |
| rw [ExternalTensor.Binomial.iteration_isZero_iff Phi derived H 0 3 (shiftedSplit Phi comparison)] |
| rw [derived_zero, iterate_val] |
|
|
| / |
| functor, once its stated stalk comparison is instantiated. -/ |
| theorem extinction (m : ℕ) [NeZero m] : |
| (∀ j < m, ¬ IsZero ((Phi.obj^[j]) (derived (test m)))) ∧ |
| IsZero ((Phi.obj^[m]) (derived (test m))) := by |
| constructor |
| · intro j hj hz |
| have hs := (isZero_iff Phi comparison j (test m)).mp hz |
| have hn := selectionIterate_test_nontrivial m j hj |
| exact @not_subsingleton _ hn hs |
| · exact (isZero_iff Phi comparison m (test m)).mpr (selectionIterate_test_subsingleton m) |
| end LittleFinitistic.SelectedIteration |
|
|
| end |
|
|
| end OAI |
|
|
