| import OAI.Algebra.Finitistic.Dimension.PDResidueScalar |
| import OAI.Algebra.Finitistic.Selected.SelectedBarResolution |
| import OAI.Algebra.Finitistic.Selected.SelectedMinimalResolution |
| import OAI.Algebra.Finitistic.Dimension.PDBarDetection |
| import OAI.Algebra.Finitistic.Dimension.PDDerivedBarConsumer |
| import OAI.Algebra.Finitistic.Dimension.PDSquareZeroResidue |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.SelectedPDDetection |
| open SelectedOrdinaryAlgebra SelectedOrdinaryModules |
| |
| instance residueModuleK : Module ℂ (PDTensor.Residue (A := A)) := PDResidueScalar.moduleK ℂ A |
| instance residueTowerK : IsScalarTower ℂ Aᵐᵒᵖ (PDTensor.Residue (A := A)) := PDResidueScalar.towerK ℂ A |
| |
| open SelectedMinimalResolution (finiteModule) |
| |
| abbrev barTensor (Y : FiniteSelectionFamily.S.{0}) |
| (U : Type) [AddCommGroup U] [Module ℂ U] [Module Aᵐᵒᵖ U] := |
| PDBar.tensorComplex (finiteModule Y) U (SelectedBarResolution.resolution Y) |
| |
| def torIsoBar (Y : FiniteSelectionFamily.S.{0}) |
| (U : Type) [AddCommGroup U] [Module ℂ U] [Module Aᵐᵒᵖ U] (n : ℕ) : |
| PDTensor.Tor (k := ℂ) U (module Y) n ≅ (barTensor Y U).homology n := |
| PDBar.torIsoResolution (finiteModule Y) U (SelectedBarResolution.resolution Y) n |
| |
| theorem pdLE_iff_residue_bar (Y : FiniteSelectionFamily.S.{0}) (n : ℕ) : |
| HasProjectiveDimensionLE (module Y) n ↔ |
| IsZero ((barTensor Y (PDTensor.Residue (A := A))).homology (n+1)) := |
| PDBar.pdLE_iff_residue_resolution (finiteModule Y) (SelectedBarResolution.resolution Y) n |
| |
| theorem pdLE_iff_simple_bar (Y : FiniteSelectionFamily.S.{0}) (n : ℕ) : |
| HasProjectiveDimensionLE (module Y) n ↔ |
| ∀ S : Submodule Aᵐᵒᵖ (PDTensor.Residue (A := A)), IsSimpleModule Aᵐᵒᵖ S → |
| IsZero ((barTensor Y S).homology (n+1)) := |
| PDBar.pdLE_iff_simple_resolution (finiteModule Y) (SelectedBarResolution.resolution Y) n |
| |
| theorem finite_pd_of_bar (Y : FiniteSelectionFamily.S.{0}) |
| (h : ∃ b : ℕ, ∀ n, b < n → |
| IsZero ((barTensor Y (PDTensor.Residue (A := A))).homology n)) : |
| projectiveDimension (module Y) < ⊤ := |
| PDTensor.finite_pd_of_resolution (finiteModule Y) (SelectedBarResolution.resolution Y) h |
| |
| theorem lower_bound_of_bar (Y : FiniteSelectionFamily.S.{0}) |
| (U : Type) [AddCommGroup U] [Module ℂ U] [Module Aᵐᵒᵖ U] |
| (r n : ℕ) (hrn : r ≤ n) (h : ¬ IsZero ((barTensor Y U).homology n)) : |
| r ≤ projectiveDimension (module Y) := |
| PDTensor.lower_bound_of_resolution (finiteModule Y) U |
| (SelectedBarResolution.resolution Y) r n hrn h |
| |
| local instance : HasDerivedCategory.{1} (ModuleCat.{0} ℂ) := |
| HasDerivedCategory.standard (ModuleCat.{0} ℂ) |
| |
| theorem finite_pd_of_derived_bar (Y : FiniteSelectionFamily.S.{0}) |
| (Z : DerivedCategory (ModuleCat.{0} ℂ)) |
| (e : PDBar.derivedObject (barTensor Y (PDTensor.Residue (A := A))) ≅ Z) |
| (h : ∃ b : ℕ, ∀ n, b < n → IsZero |
| ((DerivedCategory.homologyFunctor (ModuleCat ℂ) (-(n : ℤ))).obj Z)) : |
| projectiveDimension (module Y) < ⊤ := |
| PDBar.finite_pd_of_derived_iso (finiteModule Y) (SelectedBarResolution.resolution Y) Z e h |
| |
| theorem lower_bound_of_derived_bar (Y : FiniteSelectionFamily.S.{0}) |
| (U : Type) [AddCommGroup U] [Module ℂ U] [Module Aᵐᵒᵖ U] |
| (Z T : DerivedCategory (ModuleCat.{0} ℂ)) |
| (e : PDBar.derivedObject (barTensor Y U) ≅ Z) |
| (i : T ⟶ Z) (p : Z ⟶ T) (hip : i ≫ p = 𝟙 T) |
| (r n : ℕ) (hrn : r ≤ n) |
| (h : ¬ IsZero ((DerivedCategory.homologyFunctor (ModuleCat ℂ) (-(n : ℤ))).obj T)) : |
| r ≤ projectiveDimension (module Y) := |
| PDBar.lower_bound_of_derived_retract (finiteModule Y) (SelectedBarResolution.resolution Y) |
| U Z T e i p hip r n hrn h |
| end LittleFinitistic.SelectedPDDetection |
| |
| end |
| |
| end OAI |
| |
