| import OAI.Algebra.AuslanderReiten.Coordinates.MapExtend |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory Category Limits ZeroObject HomologicalComplex |
| namespace ArExplicit.ResolutionProducts |
| universe u v u' v' |
| variable {C : Type u} [Category.{v} C] [Abelian C] |
| variable {D : Type u'} [Category.{v'} D] [Abelian D] |
| variable (F : C ⥤ D) [F.Additive] [PreservesFiniteLimits F] [PreservesFiniteColimits F] |
| variable [F.PreservesProjectiveObjects] |
| variable {X : C} (P : ProjectiveResolution X) |
|
|
| omit [Abelian C] [Abelian D] [F.Additive] [PreservesFiniteLimits F] |
| [PreservesFiniteColimits F] [F.PreservesProjectiveObjects] in |
| private theorem map_composite_cancel {U V W Z : C} {U' Z' : D} |
| (a : U' ⟶ F.obj U) (b : U ⟶ V) (f : V ⟶ W) (e : W ≅ Z) (t : F.obj W ⟶ Z') : |
| a ≫ F.map (b ≫ f ≫ e.hom) ≫ (F.map e.inv ≫ t) = |
| (a ≫ F.map b) ≫ F.map f ≫ t := by |
| simp only [Functor.map_comp, Category.assoc] |
| rw [← F.map_comp_assoc e.hom e.inv t, e.hom_inv_id, F.map_id, id_comp] |
|
|
| def mapResolutionCochainIso : (mapResolution F P).cochainComplex ≅ |
| (F.mapHomologicalComplex (.up ℤ)).obj P.cochainComplex := |
| mapExtendIso F P.complex ComplexShape.embeddingDownNat |
|
|
| theorem mapResolutionCochainIso_coordinate (n : ℤ) (j : ℕ) (h : -(j : ℤ) = n) : |
| (mapResolutionCochainIso F P).hom.f n ≫ F.map (P.cochainComplexXIso n j h).hom = |
| ((mapResolution F P).cochainComplexXIso n j h).hom := |
| mapExtendXIso_hom_XIso F P.complex (ComplexShape.embeddingDownNat.r_eq_some h) |
|
|
| theorem mapResolution_π_zero : (mapResolution F P).π.f 0 = F.map (P.π.f 0) := by |
| change F.map (P.π.f 0) ≫ ((singleMapHomologicalComplex F (.down ℕ) 0).hom.app X).f 0 = _ |
| rw [singleMapHomologicalComplex_hom_app_self] |
| change F.map (P.π.f 0) ≫ (F.map (𝟙 X) ≫ 𝟙 (F.obj X)) = _ |
| simp |
|
|
| theorem mapResolutionCochainIso_π : |
| (mapResolutionCochainIso F P).hom ≫ |
| (F.mapHomologicalComplex (.up ℤ)).map P.π' ≫ |
| (F.mapCochainComplexSingleFunctor 0).hom.app X = (mapResolution F P).π' := by |
| apply to_single_hom_ext |
| change (mapResolutionCochainIso F P).hom.f 0 ≫ F.map (P.π'.f 0) ≫ |
| ((singleMapHomologicalComplex F (.up ℤ) 0).hom.app X).f 0 = (mapResolution F P).π'.f 0 |
| rw [P.π'_f_zero, |
| (mapResolution F P).π'_f_zero, mapResolution_π_zero, |
| singleMapHomologicalComplex_hom_app_self] |
| have he := map_composite_cancel F ((mapResolutionCochainIso F P).hom.f 0) |
| (P.cochainComplexXIso 0 0 (by simp)).hom (show P.complex.X 0 ⟶ X from P.π.f 0) |
| (singleObjXSelf (.up ℤ) 0 X).symm (singleObjXSelf (.up ℤ) 0 (F.obj X)).inv |
| exact he.trans (congrArg (fun a => a ≫ F.map (P.π.f 0) ≫ |
| (singleObjXSelf (.up ℤ) 0 (F.obj X)).inv) (mapResolutionCochainIso_coordinate F P 0 0 (by simp))) |
|
|
| end ArExplicit.ResolutionProducts |
|
|
| end |
|
|
| end OAI |
|
|
