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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | 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
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