| import OAI.Algebra.Finitistic.Dimension.PDAllSimples |
| import OAI.Algebra.Finitistic.Dimension.PDResolutionDetection |
| import OAI.Algebra.Finitistic.Bar.Bar |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.PDBar |
| universe u |
| variable {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A] |
| variable (M : PDMinimal.FiniteModule k A) |
| variable (U : Type u) [AddCommGroup U] [Module k U] [Module Aᵐᵒᵖ U] |
|
|
| abbrev tensorComplex (P : ProjectiveResolution M.cat) : ChainComplex (ModuleCat k) ℕ := |
| ((PDTensor.tensorFunctor k A U).mapHomologicalComplex _).obj P.complex |
|
|
| /-- Any actual projective resolution computes the same Tor, including the |
| constructed absolute bar resolution. -/ |
| def torIsoResolution (P : ProjectiveResolution M.cat) (n : ℕ) : |
| PDTensor.Tor (k := k) U M.cat n ≅ (tensorComplex M U P).homology n := |
| P.isoLeftDerivedObj (PDTensor.tensorFunctor k A U) n |
|
|
| abbrev resolution : ProjectiveResolution M.cat := DerivedBar.projectiveResolution k A M |
|
|
| /-- No extra identification between the bar augmentation and M is assumed. -/ |
| def torIsoBar (n : ℕ) : |
| PDTensor.Tor (k := k) U M.cat n ≅ (tensorComplex M U (resolution M)).homology n := |
| torIsoResolution M U (resolution M) n |
|
|
| variable {U} |
| theorem pdLE_iff_residue_resolution (P : ProjectiveResolution M.cat) (n : ℕ) : |
| HasProjectiveDimensionLE M.cat n ↔ |
| IsZero ((tensorComplex M (PDTensor.Residue (A := A)) P).homology (n+1)) := by |
| rw [PDTensor.pdLE_iff_residueTor_zero M n] |
| exact (torIsoResolution M _ P (n+1)).isZero_iff |
|
|
| theorem pdLE_iff_simple_resolution (P : ProjectiveResolution M.cat) (n : ℕ) : |
| HasProjectiveDimensionLE M.cat n ↔ |
| ∀ S : Submodule Aᵐᵒᵖ (PDTensor.Residue (A := A)), IsSimpleModule Aᵐᵒᵖ S → |
| IsZero ((tensorComplex M S P).homology (n+1)) := by |
| rw [PDTensor.pdLE_iff_simpleTor_zero M n] |
| exact forall_congr' fun S => imp_congr_right fun _ => |
| (torIsoResolution M S P (n+1)).isZero_iff |
| end LittleFinitistic.PDBar |
|
|
| end |
|
|
| end OAI |
|
|
