| import OAI.Algebra.AuslanderReiten.Complete.CompleteTailFunctor |
| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomAugmentedExt |
| import OAI.Algebra.AuslanderReiten.Homology.NatComparisonFunctor |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits |
| open CochainComplex CochainComplex.HomComplex |
| namespace ArExplicit.CompleteComparison |
| universe u |
| variable {k R S : Type u} [Field k] [CharP k 2] [Ring R] [Algebra k R] |
| [Ring S] [Algebra k S] |
| variable (F : ModuleCat.{u} R ⥤ ModuleCat.{u} S) |
|
|
| private theorem augmented_square {A B X : ModuleCat.{u} R} |
| {A' B' X' : ModuleCat.{u} S} (a : A' ⟶ B') (b : A ⟶ B) |
| (u : A' ⟶ F.obj A) (v : B' ⟶ F.obj B) |
| (π : B ⟶ X) (π' : B' ⟶ X') (e : X' ≅ F.obj X) |
| (hab : a ≫ v = u ≫ F.map b) (hπ : v ≫ F.map π = π' ≫ e.hom) : |
| a ≫ π' = (u ≫ F.map (b ≫ π)) ≫ e.inv := by |
| apply (cancel_mono e.hom).1 |
| simp only [Category.assoc, e.inv_hom_id, Category.comp_id] |
| rw [← hπ, ← Category.assoc, hab, Category.assoc, ← F.map_comp] |
|
|
| variable [HasDerivedCategory (ModuleCat.{u} R)] [HasDerivedCategory (ModuleCat.{u} S)] |
| variable [F.Additive] [PreservesFiniteLimits F] [PreservesFiniteColimits F] |
| [F.PreservesProjectiveObjects] |
|
|
| variable (K : CochainComplex (ModuleCat.{u} R) ℤ) |
| variable (hK : TotallyAcyclicComplex (R := R) (chainOfCochain K)) |
| variable (hFK : TotallyAcyclicComplex (R := S) (chainOfCochain (functorComplex F K))) |
| variable (e : ModuleCat.of S (Cokernel (chainOfCochain (functorComplex F K)) 0) ≅ |
| F.obj (ModuleCat.of R (Cokernel (chainOfCochain K) 0))) |
| variable (hπ : (tailFunctorHom F K).f 0 ≫ F.map (((completeExtResolution (R := R) hK)).π.f 0) = ((completeExtResolution (R := S) hFK)).π.f 0 ≫ e.hom) |
| include hπ |
|
|
| def completeTailResolutionHom : |
| ProjectiveResolution.Hom (completeExtResolution (R := S) hFK) (ResolutionProducts.mapResolution F (completeExtResolution (R := R) hK)) e.hom where |
| hom := tailFunctorHom F K |
| hom_f_zero_comp_π_f_zero := by |
| erw [ResolutionProducts.mapResolution_π_zero, ChainComplex.single₀_map_f_zero] |
| exact hπ |
|
|
| omit [HasDerivedCategory (ModuleCat.{u} R)] [HasDerivedCategory (ModuleCat.{u} S)] |
| [PreservesFiniteLimits F] [PreservesFiniteColimits F] [F.PreservesProjectiveObjects] in |
| theorem tailFunctor_augmentation (r : ℕ) (x : Cochain K K (r : ℤ)) : |
| ResolutionProducts.natComparisonAugmentation (R := S) (completeExtResolution (R := S) hFK) r (cochainTailMap (functorComplex F K) (functorComplex F K) r (x.map F)) = |
| ((tailFunctorHom F K).f r ≫ |
| F.map (ResolutionProducts.natComparisonAugmentation (R := R) (completeExtResolution (R := R) hK) r (cochainTailMap K K r x))) ≫ |
| e.inv := by |
| exact augmented_square F |
| (cochainTailMap (functorComplex F K) (functorComplex F K) r (x.map F) 0) |
| (cochainTailMap K K r x 0) ((tailFunctorHom F K).f r) ((tailFunctorHom F K).f 0) |
| ((completeExtResolution hK).π.f 0) ((completeExtResolution hFK).π.f 0) e |
| (tailFunctorHom_cochain F K r 0 x) hπ |
|
|
| theorem restrictedEndExt_functor (r : ℕ) (x : Cocycle K K (r : ℤ)) : |
| restrictedEndExt (k := k) (functorComplex F K) hFK r |
| (fullHomCocycleMap F K K (r : ℤ) x) = |
| ((Ext.mk₀ e.hom).comp |
| ((restrictedEndExt (k := k) K hK r x).mapExactFunctor F) (zero_add r)).comp |
| (Ext.mk₀ e.inv) (add_zero r) := by |
| exact ResolutionProducts.natComparisonEndExt_map (k := k) F |
| (completeExtResolution hK) (completeExtResolution hFK) e |
| (completeTailResolutionHom F K hK hFK e hπ) r |
| (cochainTailMap K K r x.val) (cochainTailMap_comm (k := k) K K r x) |
| (cochainTailMap (functorComplex F K) (functorComplex F K) r |
| (fullHomCocycleMap F K K (r : ℤ) x).val) |
| (cochainTailMap_comm (k := k) _ _ r (fullHomCocycleMap F K K (r : ℤ) x)) |
| (tailFunctor_augmentation F K hK hFK e hπ r x.val) |
|
|
| theorem fullPositiveExtEquiv_functor_mk (n : ℕ) (x : Cocycle K K ((n+1 : ℕ) : ℤ)) : |
| fullPositiveExtEquiv (k := k) (R := S) (functorComplex F K) (functorComplex F K) hFK hFK.toExactProjectiveComplex n |
| (CohomologyClass.mk (fullHomCocycleMap F K K ((n+1 : ℕ) : ℤ) x)) = |
| ((Ext.mk₀ e.hom).comp |
| ((fullPositiveExtEquiv (k := k) (R := R) K K hK hK.toExactProjectiveComplex n |
| (CohomologyClass.mk x)).mapExactFunctor F) (zero_add _)).comp |
| (Ext.mk₀ e.inv) (add_zero _) := by |
| rw [fullPositiveExtEquiv_mk, fullPositiveExtEquiv_mk] |
| exact restrictedEndExt_functor (k := k) F K hK hFK e hπ (n+1) x |
|
|
| end ArExplicit.CompleteComparison |
|
|
| end |
|
|
| end OAI |
|
|
