| import Mathlib.CategoryTheory.Abelian.Exact |
| import OAI.Algebra.Finitistic.Dimension.PDBarTensorComparison |
| import OAI.Algebra.Finitistic.Dimension.PDDerivedBarConsumer |
| import Mathlib.Algebra.Homology.ShortComplex.PreservesHomology |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.PDBarQuotient |
| universe u |
| variable {k A D : Type u} [Field k] [Ring A] [Algebra k A] [Ring D] [Algebra k D] |
| variable (f : A →ₐ[k] D) |
| variable (N : Type u) [AddCommGroup N] [Module k N] [Module A N] [IsScalarTower k A N] |
| variable [FiniteDimensional k A] [FiniteDimensional k N] |
|
|
| /-- Restriction of scalars preserves the actual homology of a D-complex. -/ |
| def restrictHomologyIso (C : ChainComplex (ModuleCat.{u} D) ℕ) (n : ℕ) : |
| (((ModuleCat.restrictScalars (algebraMap k D)).mapHomologicalComplex _).obj C).homology n ≅ |
| (ModuleCat.restrictScalars (algebraMap k D)).obj (C.homology n) := |
| by |
| let F : ModuleCat.{u} D ⥤ ModuleCat.{u} k := ModuleCat.restrictScalars (algebraMap k D) |
| letI : F.PreservesHomology := Functor.preservesHomology_of_preservesMonos_and_cokernels F |
| exact (C.sc n).mapHomologyIso F |
|
|
| /-- Actual nonzero D-module bar homology, with its exterior action intact, |
| forces the same projective-dimension lower bound over A. -/ |
| theorem lower_bound_of_base_bar (r n : ℕ) (hrn : r ≤ n) |
| (h : ¬ IsZero ((DerivedBar.twoSidedComplex k A D N f).homology n)) : |
| r ≤ projectiveDimension (ModuleCat.of A N) := by |
| apply PDTensor.lower_bound_of_nonzero_Tor (RightBase f) (PDMinimal.FiniteModule.of (k := k) (A := A) N) r n hrn |
| intro hz |
| have hz' := hz.of_iso ((torIsoTwoSided f N n) ≪≫ |
| restrictHomologyIso (DerivedBar.twoSidedComplex k A D N f) n).symm |
| apply h |
| have hs := (ModuleCat.isZero_iff_subsingleton (R := k)).mp hz' |
| exact ModuleCat.isZero_iff_subsingleton.mpr hs |
|
|
| local instance : HasDerivedCategory.{u+1} (ModuleCat.{u} D) := |
| HasDerivedCategory.standard (ModuleCat.{u} D) |
|
|
| abbrev baseDerived := DerivedCategory.Q.obj |
| ((DerivedBar.twoSidedComplex k A D N f).extend ComplexShape.embeddingDownNat) |
|
|
| def baseDerivedHomologyIso (n : ℕ) : |
| (DerivedCategory.homologyFunctor (ModuleCat D) (-(n : ℤ))).obj (baseDerived f N) ≅ |
| (DerivedBar.twoSidedComplex k A D N f).homology n := |
| (DerivedCategory.homologyFunctorFactors (ModuleCat D) (-(n : ℤ))).app _ ≪≫ |
| (DerivedBar.twoSidedComplex k A D N f).extendHomologyIso ComplexShape.embeddingDownNat rfl |
|
|
| /-- Consumer of an actual D-linear derived bar splitting. No splitting or |
| nonzero-iterate theorem is assumed implicitly; the iso, retract and class are inputs. -/ |
| theorem lower_bound_of_base_derived_retract |
| (Z T : DerivedCategory (ModuleCat.{u} D)) (e : baseDerived f N ≅ Z) |
| (i : T ⟶ Z) (p : Z ⟶ T) (hip : i ≫ p = 𝟙 T) |
| (r n : ℕ) (hrn : r ≤ n) |
| (h : ¬ IsZero ((DerivedCategory.homologyFunctor (ModuleCat D) (-(n : ℤ))).obj T)) : |
| r ≤ projectiveDimension (ModuleCat.of A N) := by |
| apply lower_bound_of_base_bar f N r n hrn |
| intro hz |
| let F := DerivedCategory.homologyFunctor (ModuleCat D) (-(n : ℤ)) |
| have hz' : IsZero (F.obj Z) := hz.of_iso (F.mapIso e.symm ≪≫ baseDerivedHomologyIso f N n) |
| apply h |
| apply (IsZero.iff_id_eq_zero _).mpr |
| have he : F.map i ≫ F.map p = 𝟙 (F.obj T) := by rw [← F.map_comp, hip, F.map_id] |
| rw [← he, hz'.eq_of_tgt (F.map i) 0, zero_comp] |
| end LittleFinitistic.PDBarQuotient |
|
|
| end |
|
|
| end OAI |
|
|
