| 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) |
|
|
| / |
| abbrev object (r : ℕ) := (chainToIntegerDerived D).obj |
| ((derivedTensorIterate k D X r).obj |
| ((chainQ D).obj ((ChainComplex.single₀ (ModuleCat D)).obj N))) |
|
|
| / |
| 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 |
| |
| / |
| 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 |
| |
