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ae00908 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | 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
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