| import OAI.Algebra.AuslanderReiten.Resolutions.MapResolutionExt |
| import OAI.Algebra.AuslanderReiten.Cocycles.CocycleFunctor |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits |
| open CochainComplex CochainComplex.HomComplex HomologicalComplex |
| namespace ArExplicit.ResolutionProducts |
| universe u v u' v' |
| variable {C : Type u} [Category.{v} C] [Abelian C] [HasExt.{v} C] |
| variable {D : Type u'} [Category.{v'} D] [Abelian D] [HasExt.{v'} D] |
| variable (F : C ⥤ D) [F.Additive] [PreservesFiniteLimits F] |
| [PreservesFiniteColimits F] [F.PreservesProjectiveObjects] |
| variable {X Y : C} (P : ProjectiveResolution X) {n : ℕ} |
|
|
| omit [Abelian C] [Abelian D] [HasExt.{v} C] [HasExt.{v'} D] [F.Additive] |
| [PreservesFiniteLimits F] [PreservesFiniteColimits F] [F.PreservesProjectiveObjects] in |
| private theorem map_eval_cancel {U V W Z : C} {U' Z' : D} |
| (a : U' ⟶ F.obj U) (b : U ⟶ V) (f : V ⟶ W) (e : Z ≅ W) (t : Z' ≅ F.obj W) : |
| (a ≫ F.map ((b ≫ f) ≫ e.inv) ≫ (F.map e.hom ≫ t.inv)) ≫ t.hom = |
| (a ≫ F.map b) ≫ F.map f := by |
| simp only [Functor.map_comp, Category.assoc, t.inv_hom_id, Category.comp_id] |
| rw [← F.map_comp e.inv e.hom, e.inv_hom_id, F.map_id, Category.comp_id] |
|
|
| omit [HasExt.{v} C] [HasExt.{v'} D] in |
| theorem mappedResolutionCocycle_eq |
| (x : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ)) : |
| mappedResolutionCocycle F P x = |
| ((fullHomCocycleMap F _ _ _ x).precomp (mapResolutionCochainIso F P).hom).postcomp |
| ((F.mapCochainComplexSingleFunctor 0).hom.app Y) := by |
| apply Cocycle.equivHomShift.symm.injective |
| rw [mappedResolutionCocycle, AddEquiv.symm_apply_apply, |
| Cocycle.equivHomShift_symm_postcomp, Cocycle.equivHomShift_symm_precomp, cocycle_map_hom] |
| exact (Category.assoc _ _ _).symm |
|
|
| omit [HasExt.{v} C] [HasExt.{v'} D] in |
| theorem mappedResolutionCocycle_v |
| (x : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ)) |
| (i j : ℤ) (hij : i+n=j) : |
| (mappedResolutionCocycle F P x).val.v i j hij = |
| (mapResolutionCochainIso F P).hom.f i ≫ F.map (x.val.v i j hij) ≫ |
| ((F.mapCochainComplexSingleFunctor 0).hom.app Y).f j := by |
| rw [mappedResolutionCocycle_eq] |
| change (((Cochain.ofHom _).comp (x.val.map F) (zero_add (n : ℤ))).comp |
| (Cochain.ofHom _) (add_zero (n : ℤ))).v i j hij = _ |
| simp only [Cochain.comp_zero_cochain_v, Cochain.zero_cochain_comp_v, Cochain.ofHom_v, |
| Cochain.map_v] |
| exact Category.assoc _ _ _ |
|
|
| theorem map_extMk (f : P.complex.X n ⟶ Y) |
| (hf : P.complex.d (n+1) n ≫ f = 0) : |
| (P.extMk f (n+1) rfl hf).mapExactFunctor F = |
| (mapResolution F P).extMk (F.map f) (n+1) rfl |
| (by change F.map (P.complex.d (n+1) n) ≫ F.map f = 0 |
| rw [← F.map_comp, hf, F.map_zero]) := by |
| unfold ProjectiveResolution.extMk |
| rw [map_extOfCocycle] |
| apply congrArg (mapResolution F P).extEquivCohomologyClass.symm |
| apply congrArg CohomologyClass.mk |
| apply Cocycle.ext |
| apply (Cochain.toSingleEquiv (p := -(n : ℤ)) (q := 0) (n := (n : ℤ)) (by omega)).injective |
| rw [Cocycle.toSingleMk_coe, Cochain.toSingleEquiv_toSingleMk] |
| change (mappedResolutionCocycle F P _).val.v (-(n : ℤ)) 0 (by omega) ≫ |
| (singleObjXSelf (.up ℤ) 0 (F.obj Y)).hom = _ |
| rw [mappedResolutionCocycle_v, Cocycle.toSingleMk_coe, Cochain.toSingleMk_v] |
| change ((mapResolutionCochainIso F P).hom.f (-(n : ℤ)) ≫ |
| F.map (((P.cochainComplexXIso (-(n : ℤ)) n rfl).hom ≫ f) ≫ |
| (singleObjXSelf (.up ℤ) 0 Y).inv) ≫ |
| ((singleMapHomologicalComplex F (.up ℤ) 0).hom.app Y).f 0) ≫ |
| (singleObjXSelf (.up ℤ) 0 (F.obj Y)).hom = _ |
| rw [singleMapHomologicalComplex_hom_app_self] |
| exact (map_eval_cancel F ((mapResolutionCochainIso F P).hom.f (-(n : ℤ))) |
| (P.cochainComplexXIso (-(n : ℤ)) n rfl).hom f |
| (singleObjXSelf (.up ℤ) 0 Y) (singleObjXSelf (.up ℤ) 0 (F.obj Y))).trans |
| (congrArg (fun g => g ≫ F.map f) (mapResolutionCochainIso_coordinate F P _ n rfl)) |
|
|
| end ArExplicit.ResolutionProducts |
|
|
| end |
|
|
| end OAI |
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|
