| import OAI.Algebra.Finitistic.Bar.SplitProjectiveTail | |
| import OAI.Algebra.Finitistic.Iteration.FiniteIntegerIterate | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex | |
| 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] | |
| [IsScalarTower k D X] | |
| local instance : HasDerivedCategory.{u+1} (ModuleCat.{u} D) := | |
| HasDerivedCategory.standard (ModuleCat.{u} D) | |
| variable (g : ℕ) (hg : ∀ M : ModuleCat.{u} D, HasProjectiveDimensionLE M g) | |
| variable (N : ModuleCat.{u} D) | |
| abbrev resolvedModel (r : ℕ) := (tensorTotalIterate k D X r).obj | |
| ((ChainComplex.single₀ (ModuleCat D)).obj N) | |
| include hg in | |
| /-- A bounded input remains cohomologically bounded after actual derived tensor. | |
| The input is resolved by its genuine projective bar total. -/ | |
| theorem tensor_exact_above (Z : ChainComplex (ModuleCat.{u} D) ℕ) (m : ℕ) | |
| (hZ : ∀ i, m < i → Z.ExactAt i) (n : ℕ) (hn : m+g+2 < n) : | |
| ((tensorTotalFunctor k D X).obj Z).ExactAt n := by | |
| apply ProjectiveTail.map_exact_tail (barTotal k D Z) (BimoduleTensor.tensorFunctor k D X) | |
| (m+1) g | |
| · intro i hi | |
| exact (exactAt_iff_of_quasiIsoAt (barTotalAugmentation k D Z) i).mpr (hZ i (by omega)) | |
| · exact barTotal_projective k D Z | |
| · exact hg | |
| · omega | |
| include hg in | |
| /-- The actual finite iteration has an explicit finite homological support bound. | |
| This bound has a two-degree allowance per step; no optimal upper bound is claimed. -/ | |
| theorem resolvedModel_exact_above (r n : ℕ) (hn : r*(g+2) < n) : | |
| (resolvedModel k D X N r).ExactAt n := by | |
| induction r generalizing n with | |
| | zero => | |
| cases n with | |
| | zero => simp at hn | |
| | succ n => exact ChainComplex.exactAt_succ_single_obj _ _ | |
| | succ r ih => | |
| apply tensor_exact_above k D X g hg (resolvedModel k D X N r) (r*(g+2)) ih n | |
| simpa only [Nat.add_mul, Nat.one_mul, Nat.add_assoc] using hn | |
| def resolvedModelIso (r : ℕ) : object k D X N r ≅ | |
| DerivedCategory.Q.obj ((integerExtension D).obj (resolvedModel k D X N r)) := | |
| (chainToIntegerDerived D).mapIso | |
| ((derivedTensorIterateIso k D X r).app ((ChainComplex.single₀ (ModuleCat D)).obj N)) ≪≫ | |
| (chainToIntegerDerivedIso D).app (resolvedModel k D X N r) | |
| include hg in | |
| /-- Actual derived tensor iterates have finite lower cohomological support over | |
| any ring with the supplied genuine global projective-dimension bound. -/ | |
| theorem isGE_bound (r : ℕ) : (object k D X N r).IsGE (-((r*(g+2) : ℕ) : ℤ)) := by | |
| apply (DerivedCategory.isGE_iff _ _).mpr | |
| intro i hi | |
| obtain ⟨n,rfl⟩ := Int.exists_eq_neg_ofNat (show i ≤ 0 by omega) | |
| let e := (DerivedCategory.homologyFunctor (ModuleCat D) (-(n : ℤ))).mapIso | |
| (resolvedModelIso k D X N r) ≪≫ | |
| (DerivedCategory.homologyFunctorFactors (ModuleCat D) (-(n : ℤ))).app | |
| ((integerExtension D).obj (resolvedModel k D X N r)) ≪≫ | |
| integerExtensionHomologyIso D (resolvedModel k D X N r) n | |
| exact e.isZero_iff.mpr ((resolvedModel_exact_above k D X g hg N r n (by omega)).isZero_homology) | |
| variable [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k Dᵐᵒᵖ X] | |
| include hg in | |
| theorem bounded_amplitude (r : ℕ) : | |
| (object k D X N r).IsGE (-((r*(g+2) : ℕ) : ℤ)) ∧ (object k D X N r).IsLE 0 := | |
| ⟨isGE_bound k D X g hg N r, inferInstance⟩ | |
| end DerivedBar.IterateSupport | |
| end | |
| end OAI | |