| import OAI.Algebra.Finitistic.Selected.SelectedOrdinaryAlgebra |
| import OAI.Algebra.Finitistic.Selected.SelectedDerivedFunctor |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryShiftDerivedTensor |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| attribute [local instance] HasDerivedCategory.standard HasExt.standard |
| namespace LittleFinitistic.SelectedShiftedRealization |
| open SelectedFixedDiagram QuadraticEncoding SelectedOrdinaryAlgebra |
| attribute [local instance] preservesBinaryBiproducts_of_preservesBiproducts |
|
|
| / |
| abbrev shifted := SelectedRectification.bimodule⟦b⟧ |
|
|
| def model : BimoduleTotal.BoundedProjective ℂ (B P) shifted where |
| lower := -(length : ℤ) |
| upper := 0 |
| bounded := OrdinarySimulation.shifted_support _ b length a endpoints support |
| projective n := by |
| exact SelectedOrdinaryAlgebra.term_projective (n+b) |
|
|
| abbrev G := model.derived |
|
|
| / |
| verified full B-linear tensor complex. -/ |
| def comparison (Y : FiniteSelectionFamily.S.{0}) : |
| G.obj (FiniteSelectionFamily.derived Y) ≅ |
| (SelectedTensorSplitting.object (FiniteSelectionFamily.representation Y).Evaluation)⟦b⟧ := |
| model.nativeSingleEvaluation (FiniteSelectionFamily.representation Y).Evaluation ≪≫ |
| DerivedCategory.Q.mapIso |
| ((((BimoduleBalancedTensor.functor ℂ (B P) |
| (FiniteSelectionFamily.representation Y).Evaluation).mapHomologicalComplex (.up ℤ)).commShiftIso b).app |
| SelectedRectification.bimodule) ≪≫ |
| (DerivedCategory.Q.commShiftIso b).app _ |
|
|
| / |
| def realizationNested (Y : FiniteSelectionFamily.S.{0}) : |
| G.obj (FiniteSelectionFamily.derived Y) ≅ |
| (FiniteSelectionFamily.derived (FiniteSelectionFamily.H Y))⟦b⟧ ⊞ |
| ((FiniteSelectionFamily.derived (FiniteSelectionFamily.H Y))⟦(3 : ℤ)⟧)⟦b⟧ := |
| comparison Y ≪≫ (shiftFunctor _ b).mapIso (FiniteSelectionFamily.realization Y) ≪≫ |
| (shiftFunctor _ b).mapBiprod _ _ |
|
|
| / |
| def realization (Y : FiniteSelectionFamily.S.{0}) : |
| G.obj (FiniteSelectionFamily.derived Y) ≅ |
| (FiniteSelectionFamily.derived (FiniteSelectionFamily.H Y))⟦b⟧ ⊞ |
| (FiniteSelectionFamily.derived (FiniteSelectionFamily.H Y))⟦b+3⟧ := |
| realizationNested Y ≪≫ biprod.mapIso (Iso.refl _) |
| ((shiftAdd _ (3 : ℤ) b).symm ≪≫ eqToIso (by rw [add_comm (3 : ℤ) b])) |
|
|
| def iteration (j : ℕ) (Y : FiniteSelectionFamily.S.{0}) : |
| (G.obj^[j]) (FiniteSelectionFamily.derived Y) ≅ |
| ExternalTensor.Binomial.binomialObject b 3 j |
| (FiniteSelectionFamily.derived ((FiniteSelectionFamily.H^[j]) Y)) := |
| ExternalTensor.Binomial.binomialIteration G FiniteSelectionFamily.derived |
| FiniteSelectionFamily.H b 3 realization j Y |
|
|
| theorem isZero_iff (j : ℕ) (Y : FiniteSelectionFamily.S.{0}) : |
| IsZero ((G.obj^[j]) (FiniteSelectionFamily.derived Y)) ↔ |
| Subsingleton (selectionIterate Y.val j) := |
| (ExternalTensor.Binomial.iteration_isZero_iff G FiniteSelectionFamily.derived |
| FiniteSelectionFamily.H b 3 realization j Y).trans |
| ((FiniteSelectionFamily.derived_zero _).trans |
| (by rw [FiniteSelectionFamily.iterate_val])) |
|
|
| theorem extinction (m : ℕ) [NeZero m] : |
| (∀ j < m, ¬ IsZero ((G.obj^[j]) (FiniteSelectionFamily.derived (FiniteSelectionFamily.test m)))) ∧ |
| IsZero ((G.obj^[m]) (FiniteSelectionFamily.derived (FiniteSelectionFamily.test m))) := by |
| constructor |
| · intro j hj hz |
| have hs := (isZero_iff j (FiniteSelectionFamily.test m)).mp hz |
| exact @not_subsingleton _ (selectionIterate_test_nontrivial m j hj) hs |
| · exact (isZero_iff m (FiniteSelectionFamily.test m)).mpr (selectionIterate_test_subsingleton m) |
| end LittleFinitistic.SelectedShiftedRealization |
|
|
| end |
|
|
| end OAI |
|
|
