| import OAI.Algebra.Finitistic.Iteration.IntegerIterateAmplitude |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| namespace DerivedBar.IterateSupport |
| universe u w v |
| variable (k D : Type u) [Field k] [Ring D] [Algebra k D] |
| variable {E : Type w} [Category.{v} E] [Abelian E] |
| variable (F : ModuleCat.{u} D ⥤ E) [F.Additive] |
| variable (g : ℕ) (hg : ∀ M : ModuleCat.{u} D, HasProjectiveDimensionLE M g) |
|
|
| include hg in |
| /-- A terminal additive functor applied to the actual projective bar resolution |
| has finite support when the input has finite cohomological support. This covers |
| balanced tensor with arbitrary right modules, not just the fixed bimodule. -/ |
| theorem additive_resolved_exact_above (Z : ChainComplex (ModuleCat.{u} D) ℕ) (m : ℕ) |
| (hZ : ∀ i, m < i → Z.ExactAt i) (n : ℕ) (hn : m+g+2 < n) : |
| ((F.mapHomologicalComplex (.down ℕ)).obj (barTotal k D Z)).ExactAt n := by |
| apply ProjectiveTail.map_exact_tail (barTotal k D Z) F (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 |
|
|
| variable (X : Type u) [AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X] |
| [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X] |
| variable (N : ModuleCat.{u} D) |
|
|
| include hg in |
| omit [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k Dᵐᵒᵖ X] in |
| /-- The extra resolved tensor used for residue/base Tor also has a finite |
| bound, uniformly after every actual derived iteration. -/ |
| theorem iterated_additive_exact_above (r n : ℕ) (hn : (r+1)*(g+2) < n) : |
| ((F.mapHomologicalComplex (.down ℕ)).obj |
| (barTotal k D (resolvedModel k D X N r))).ExactAt n := by |
| apply additive_resolved_exact_above k D F g hg (resolvedModel k D X N r) |
| (r*(g+2)) (resolvedModel_exact_above k D X g hg N r) n |
| simpa only [Nat.add_mul, Nat.one_mul, Nat.add_assoc] using hn |
|
|
| local instance : HasDerivedCategory.{u+1} (ModuleCat.{u} D) := |
| HasDerivedCategory.standard (ModuleCat.{u} D) |
|
|
| include hg in |
| /-- Nonzero genuine iterates are detected in a proved finite cohomological window. -/ |
| theorem nonzero_in_finite_window (r : ℕ) (h : ¬ IsZero (object k D X N r)) : |
| ∃ n : ℕ, n ≤ r*(g+2) ∧ |
| ¬ IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) (-(n : ℤ))).obj |
| (object k D X N r)) := by |
| obtain ⟨n,hn⟩ := nonzero_homology k D X N r h |
| refine ⟨n, ?_, hn⟩ |
| by_contra hbound |
| have := isGE_bound k D X g hg N r |
| apply hn |
| exact DerivedCategory.isZero_of_isGE _ (-((r*(g+2) : ℕ) : ℤ)) (-(n : ℤ)) (by omega) |
| end DerivedBar.IterateSupport |
|
|
| end |
|
|
| end OAI |
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|
