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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 | import OAI.Algebra.Finitistic.Bar.IntegerDerivedMultibar
import Mathlib.Algebra.Homology.DerivedCategory.TStructure
import Mathlib.Algebra.Homology.Embedding.CochainComplex
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
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 : HasDerivedCategory.{u+1} (ModuleCat.{u} D) :=
HasDerivedCategory.standard (ModuleCat.{u} D)
variable (N : ModuleCat.{u} D)
/-- The already constructed genuine derived iterate, viewed in integer degrees. -/
abbrev object (r : ℕ) := (chainToIntegerDerived D).obj
((derivedTensorIterate k D X r).obj
((chainQ D).obj ((ChainComplex.single₀ (ModuleCat D)).obj N)))
/-- Every actual iterate is cohomologically nonpositive. This uses the proved
integer comparison and its actual nonnegative chain representative. -/
instance isLE_zero (r : ℕ) : (object k D X N r).IsLE 0 := by
let Z := RecursiveTotal.indexedMultibar k D X N r
have : ((integerExtension D).obj Z).IsStrictlyLE 0 :=
inferInstanceAs (CochainComplex.IsStrictlyLE (Z.extend ComplexShape.embeddingDownNat) 0)
have : (DerivedCategory.Q.obj ((integerExtension D).obj Z)).IsLE 0 := inferInstance
exact DerivedCategory.TStructure.t.isLE_of_iso
(integerDerivedIterateIndexedIso D k X N r).symm 0
theorem positive_homology_zero (r : ℕ) (i : ℤ) (hi : 0 < i) :
IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) i).obj (object k D X N r)) :=
DerivedCategory.isZero_of_isLE _ 0 i hi
/-- Nonzero actual iterates have a nonzero cohomology group in nonpositive degree. -/
theorem nonzero_homology (r : ℕ) (h : ¬ IsZero (object k D X N r)) :
∃ n : ℕ, ¬ IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) (-(n : ℤ))).obj
(object k D X N r)) := by
by_contra hn
push Not at hn
have hz (i : ℤ) : IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) i).obj
(object k D X N r)) := by
by_cases hi : i ≤ 0
· obtain ⟨n,rfl⟩ := Int.exists_eq_neg_ofNat hi
exact hn n
· exact positive_homology_zero k D X N r i (by omega)
have : (object k D X N r).IsLE (-1) :=
(DerivedCategory.isLE_iff _ _).mpr (fun i _ => hz i)
have : (object k D X N r).IsGE 0 :=
(DerivedCategory.isGE_iff _ _).mpr (fun i _ => hz i)
apply h
apply (IsZero.iff_id_eq_zero _).mpr
exact DerivedCategory.TStructure.t.zero (𝟙 _) (-1) 0 (by omega)
end DerivedBar.IterateSupport
end
end OAI
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