| import OAI.Algebra.AuslanderReiten.Resolutions.ResolutionProductCompatibility |
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| namespace OAI |
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| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
| open CochainComplex CochainComplex.HomComplex |
| namespace ArExplicit.ResolutionProducts |
| universe u v |
| variable {C : Type u} [Category.{v} C] [Abelian C] |
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|
| theorem cocycleHom_ofHom {K L : CochainComplex C ℤ} (f : K ⟶ L) : |
| Cocycle.equivHomShift.symm (Cocycle.ofHom f) = ShiftedHom.mk₀ 0 rfl f := by |
| apply HomologicalComplex.Hom.ext |
| funext i |
| simp only [CompleteComparison.cocycleHom_component, Cocycle.ofHom_coe] |
| simp only [ShiftedHom.mk₀, HomologicalComplex.comp_f, CategoryTheory.shiftFunctorZero', |
| eqToIso_refl, Iso.trans_inv, Iso.refl_inv, |
| CochainComplex.shiftFunctorZero_inv_app_f, Category.comp_id] |
| change (Cochain.ofHom f).v i (i+0) rfl = |
| f.f i ≫ (L.XIsoOfEq (show i=i+0 by omega)).hom |
| simpa only [Cochain.ofHom_v] using |
| (Cochain.v_comp_XIsoOfEq_hom (Cochain.ofHom f) i i (i+0) (add_zero i) (by omega)).symm |
|
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| variable [HasDerivedCategory C] [HasExt.{v} C] |
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| theorem endExt_identity {X : C} (P : ProjectiveResolution X) : |
| endExt P (Cocycle.ofHom (𝟙 P.cochainComplex)) = Ext.mk₀ (𝟙 X) := by |
| apply Ext.homEquiv.injective |
| change (endExt P (Cocycle.ofHom (𝟙 P.cochainComplex))).hom = (Ext.mk₀ (𝟙 X)).hom |
| rw [endExt_hom, cocycleHom_ofHom, Ext.mk₀_hom] |
| erw [ShiftedHom.map_mk₀] |
| simp [conjugate, ShiftedHom.mk₀, ← NatTrans.naturality] |
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|
| theorem extMk_augmentation {X : C} (P : ProjectiveResolution X) : |
| P.extMk (Y := X) (P.π.f 0) 1 rfl P.complex_d_comp_π_f_zero = Ext.mk₀ (𝟙 X) := by |
| rw [← endExt_identity P, endExt_eq_augmented] |
| unfold ProjectiveResolution.extMk |
| apply congrArg P.extEquivCohomologyClass.symm |
| apply congrArg CohomologyClass.mk |
| apply Subtype.ext |
| apply (Cochain.toSingleEquiv (show (0 : ℤ)+0=0 by omega)).injective |
| change (Cochain.toSingleMk ((P.cochainComplexXIso 0 0 rfl).hom ≫ P.π.f 0) |
| (by omega : (0 : ℤ)+0=0)).v 0 0 (by omega) ≫ |
| (HomologicalComplex.singleObjXSelf (.up ℤ) 0 X).hom = |
| ((Cochain.ofHom (𝟙 P.cochainComplex)).comp (Cochain.ofHom P.π') (add_zero 0)).v 0 0 |
| (by omega) ≫ (HomologicalComplex.singleObjXSelf (.up ℤ) 0 X).hom |
| rw [Cochain.id_comp, Cochain.ofHom_v, Cochain.toSingleMk_v, P.π'_f_zero] |
| dsimp only [ChainComplex.single₀_obj_zero] |
| exact congrArg (fun morphism => |
| morphism ≫ (HomologicalComplex.singleObjXSelf (.up ℤ) 0 X).hom) |
| (Category.assoc (P.cochainComplexXIso 0 0 rfl).hom (P.π.f 0) |
| (HomologicalComplex.singleObjXSelf (.up ℤ) 0 X).inv) |
|
|
| end ArExplicit.ResolutionProducts |
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| end |
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| end OAI |
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