| import OAI.Algebra.Finitistic.Iteration.IntegerIterateAmplitude |
| import OAI.Algebra.Finitistic.Bar.BarAdditive |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex ZeroObject |
| namespace DerivedBar.IterateSupport |
| universe u |
| variable (k D X : Type u) [Field k] [Ring D] [Algebra k D] |
| [AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X] |
| [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X] |
| local instance integerIterateExtinctionHasDerivedCategory : HasDerivedCategory.{u+1} (ModuleCat.{u} D) := |
| HasDerivedCategory.standard (ModuleCat.{u} D) |
|
|
| /-- The integer derived object of an actual nonnegative chain complex vanishes |
| exactly when the original chain complex is exact in every degree. -/ |
| theorem integerQ_isZero_iff (Z : ChainComplex (ModuleCat.{u} D) ℕ) : |
| IsZero ((integerQ D).obj Z) ↔ ∀ n, Z.ExactAt n := by |
| constructor |
| · intro h n |
| apply (exactAt_iff_isZero_homology _ _).mpr |
| have hz := (DerivedCategory.homologyFunctor (ModuleCat D) (-(n : ℤ))).map_isZero h |
| exact ((DerivedCategory.homologyFunctorFactors (ModuleCat D) (-(n : ℤ))).app |
| ((integerExtension D).obj Z) ≪≫ integerExtensionHomologyIso D Z n).isZero_iff.mp hz |
| · intro h |
| have : ((integerExtension D).obj Z).IsStrictlyLE 0 := by |
| exact inferInstanceAs (CochainComplex.IsStrictlyLE |
| (Z.extend ComplexShape.embeddingDownNat) 0) |
| have : ((integerQ D).obj Z).IsLE 0 := by |
| change (DerivedCategory.Q.obj ((integerExtension D).obj Z)).IsLE 0 |
| infer_instance |
| have hz (i : ℤ) : IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) i).obj |
| ((integerQ D).obj Z)) := by |
| by_cases hi : i ≤ 0 |
| · obtain ⟨n,rfl⟩ := Int.exists_eq_neg_ofNat hi |
| exact ((DerivedCategory.homologyFunctorFactors (ModuleCat D) (-(n : ℤ))).app |
| ((integerExtension D).obj Z) ≪≫ integerExtensionHomologyIso D Z n).isZero_iff.mpr |
| ((exactAt_iff_isZero_homology _ _).mp (h n)) |
| · exact DerivedCategory.isZero_of_isLE _ 0 i (by omega) |
| have : ((integerQ D).obj Z).IsLE (-1) := |
| (DerivedCategory.isLE_iff _ _).mpr (fun i _ => hz i) |
| have : ((integerQ D).obj Z).IsGE 0 := |
| (DerivedCategory.isGE_iff _ _).mpr (fun i _ => hz i) |
| apply (IsZero.iff_id_eq_zero _).mpr |
| exact DerivedCategory.TStructure.t.zero (𝟙 _) (-1) 0 (by omega) |
|
|
| local instance : (barTotalFunctor k D).PreservesZeroMorphisms where |
| map_zero Z W := by |
| apply HomologicalComplex.Hom.ext |
| funext n |
| apply HomologicalComplex₂.total.hom_ext (K := barBicomplex k D Z) |
| intro i q h |
| erw [ι_barTotalFunctor_map k D (0 : Z ⟶ W) i q n h] |
| simp [termMap_zero] |
| erw [zero_comp,comp_zero] |
| rfl |
| local instance : (tensorTotalFunctor k D X).PreservesZeroMorphisms := by |
| dsimp [tensorTotalFunctor] |
| infer_instance |
|
|
| omit [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k Dᵐᵒᵖ X] |
| /-- The actual resolve-and-tensor functor sends an acyclic chain input to an acyclic output. -/ |
| theorem tensor_acyclic (Z : ChainComplex (ModuleCat.{u} D) ℕ) |
| (h : ∀ n, Z.ExactAt n) : ∀ n, ((tensorTotalFunctor k D X).obj Z).ExactAt n := by |
| let f : Z ⟶ 0 := 0 |
| have : QuasiIso f := ⟨fun n => (quasiIsoAt_iff_exactAt f n (h n)).mpr |
| ((exactAt_iff_isZero_homology _ _).mpr |
| ((HomologicalComplex.homologyFunctor _ (.down ℕ) n).map_isZero (Limits.isZero_zero _)))⟩ |
| let G := tensorTotalFunctor k D X |
| have hz := G.map_isZero (Limits.isZero_zero (ChainComplex (ModuleCat.{u} D) ℕ)) |
| intro n |
| apply (exactAt_iff_of_quasiIsoAt (G.map f) n).mpr |
| exact (exactAt_iff_isZero_homology _ _).mpr |
| ((HomologicalComplex.homologyFunctor _ (.down ℕ) n).map_isZero hz) |
|
|
| variable (N : ModuleCat.{u} D) |
|
|
| /-- Extinction is permanent for the genuine bar-derived iteration. -/ |
| theorem zero_succ (r : ℕ) (h : IsZero (object k D X N r)) : |
| IsZero (object k D X N (r+1)) := by |
| apply (resolvedModelIso k D X N (r+1)).isZero_iff.mpr |
| apply (integerQ_isZero_iff (D := D) (resolvedModel k D X N (r+1))).mpr |
| apply tensor_acyclic k D X (resolvedModel k D X N r) |
| exact (integerQ_isZero_iff (D := D) (resolvedModel k D X N r)).mp |
| ((resolvedModelIso k D X N r).isZero_iff.mp h) |
|
|
| /-- A proved zero iterate kills every later iterate, without a native-functor comparison premise. -/ |
| theorem zero_of_le (t r : ℕ) (htr : t ≤ r) (h : IsZero (object k D X N t)) : |
| IsZero (object k D X N r) := by |
| induction r, htr using Nat.le_induction with |
| | base => exact h |
| | succ r _ ih => exact zero_succ k D X N r ih |
| end DerivedBar.IterateSupport |
|
|
| end |
|
|
| end OAI |
|
|
