| import OAI.Algebra.Finitistic.Bar.CountHomologyDetection |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits TrivSqZeroExt |
| noncomputable section |
| namespace DerivedBar.ParityBridge |
| 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 (V : ModuleCat D) : Module k V := BimoduleTensor.scalarModule k D V |
| local instance (V : ModuleCat D) : IsScalarTower k D V := |
| IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| variable (N : ModuleCat D) |
| local notation "A" => TrivSqZeroExt D X |
| local instance : Module A N := inflatedModule D X N |
| local instance : IsScalarTower k A N := IsScalarTower.of_algebraMap_smul |
| (fun a v => IsScalarTower.algebraMap_smul D a v) |
| local instance : HasDerivedCategory.{u+1} (ModuleCat.{u} D) := |
| HasDerivedCategory.standard (ModuleCat.{u} D) |
| local notation "S" => SquareZero.twoSidedWeightComplex k D X N (inflated_ideal_zero D X N) |
| local notation "T" => tensorBarComplex k A D N (fstHom k D X) |
|
|
| /-- Exact degreewise vanishing criterion for actual base-change Tor, in terms |
| of the original integer-derived count objects. -/ |
| theorem baseChangeTor_zero_iff_counts (n : ℕ) : |
| IsZero (((Balanced.tensorFunctor (fstHom k D X)).leftDerived n).obj (ModuleCat.of A N)) ↔ |
| ∀ r, IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) (-(n:ℤ))).obj |
| ((integerQ D).obj (S r))) := by |
| constructor |
| · intro hz r |
| by_contra hr |
| exact baseChangeTor_nonzero_of_count k D X N r n hr hz |
| · exact baseChangeTor_zero_of_counts k D X N n |
|
|
| /-- A uniform upper bound on the original count homologies is a uniform bound |
| on actual base-change Tor. This is the consumer for Native's uniform count window. -/ |
| theorem baseChangeTor_bounded_of_counts (B : ℕ) |
| (h : ∀ r n, B < n → |
| IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) (-(n:ℤ))).obj |
| ((integerQ D).obj (S r)))) : |
| ∀ n, B < n → |
| IsZero (((Balanced.tensorFunctor (fstHom k D X)).leftDerived n).obj (ModuleCat.of A N)) := |
| fun n hn => baseChangeTor_zero_of_counts k D X N n (fun r => h r n hn) |
|
|
| /-- Any nonzero count homology witness above r gives a nonzero actual Tor |
| witness above r, without replacing projective dimension by an invented invariant. -/ |
| theorem baseChangeTor_nonzero_ge_of_count (r : ℕ) |
| (h : ∃ n, r ≤ n ∧ |
| ¬ IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) (-(n:ℤ))).obj |
| ((integerQ D).obj (S r)))) : |
| ∃ n, r ≤ n ∧ |
| ¬ IsZero (((Balanced.tensorFunctor (fstHom k D X)).leftDerived n).obj (ModuleCat.of A N)) := by |
| obtain ⟨n, hn, hnonzero⟩ := h |
| exact ⟨n, hn, baseChangeTor_nonzero_of_count k D X N r n hnonzero⟩ |
| end DerivedBar.ParityBridge |
|
|
| end |
|
|
| end OAI |
|
|
