| import OAI.Algebra.AuslanderReiten.Resolutions.MapResolutionCochain |
| import OAI.Algebra.AuslanderReiten.Resolutions.ResolutionExtAction |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Localization |
| 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] [CategoryTheory.Limits.PreservesFiniteLimits F] |
| [CategoryTheory.Limits.PreservesFiniteColimits F] [F.PreservesProjectiveObjects] |
| variable {X Y : C} (P : ProjectiveResolution X) {n : ℕ} |
|
|
| def mappedResolutionCocycle |
| (x : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ)) : |
| Cocycle (mapResolution F P).cochainComplex |
| ((CochainComplex.singleFunctor D 0).obj (F.obj Y)) (n : ℤ) := |
| Cocycle.equivHomShift ((mapResolutionCochainIso F P).hom ≫ |
| ShiftedHom.map (Cocycle.equivHomShift.symm x) (F.mapHomologicalComplex (.up ℤ)) ≫ |
| ((F.mapCochainComplexSingleFunctor 0).hom.app Y)⟦(n : ℤ)⟧') |
|
|
| theorem precomp_extOfCocycle |
| (x : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ)) : |
| (SmallShiftedHom.mk₀ (quasiIso C (.up ℤ)) 0 rfl P.π').comp |
| (P.extEquivCohomologyClass.symm (CohomologyClass.mk x)) (add_zero (n : ℤ)) = |
| SmallShiftedHom.mk (quasiIso C (.up ℤ)) (Cocycle.equivHomShift.symm x) := by |
| have hπ : quasiIso C (.up ℤ) P.π' := by rw [mem_quasiIso_iff]; infer_instance |
| change (SmallShiftedHom.precompEquiv P.π' hπ) |
| ((SmallShiftedHom.precompEquiv P.π' hπ).symm |
| (CohomologyClass.equivOfIsKProjective (CohomologyClass.mk x))) = _ |
| rw [Equiv.apply_symm_apply] |
| rfl |
| |
| theorem map_extOfCocycle |
| (x : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ)) : |
| (P.extEquivCohomologyClass.symm (CohomologyClass.mk x)).mapExactFunctor F = |
| (mapResolution F P).extEquivCohomologyClass.symm |
| (CohomologyClass.mk (mappedResolutionCocycle F P x)) := by |
| let Φ := F.mapHomologicalComplexUpToQuasiIsoLocalizerMorphism (.up ℤ) |
| let eP : (F.mapHomologicalComplex (.up ℤ)).obj P.cochainComplex ≅ |
| (mapResolution F P).cochainComplex := (mapResolutionCochainIso F P).symm |
| let eX : (F.mapHomologicalComplex (.up ℤ)).obj ((CochainComplex.singleFunctor C 0).obj X) ≅ |
| (CochainComplex.singleFunctor D 0).obj (F.obj X) := (F.mapCochainComplexSingleFunctor 0).app X |
| let eY : (F.mapHomologicalComplex (.up ℤ)).obj ((CochainComplex.singleFunctor C 0).obj Y) ≅ |
| (CochainComplex.singleFunctor D 0).obj (F.obj Y) := (F.mapCochainComplexSingleFunctor 0).app Y |
| have h := congrArg (fun z => Φ.smallShiftedHomMap eP eY z) (precomp_extOfCocycle P x) |
| erw [Φ.smallShiftedHomMap_comp eP eX eY, |
| Φ.smallShiftedHomMap_mk₀ eP eX, Φ.smallShiftedHomMap_mk eP eY] at h |
| have hπ : eP.inv ≫ Φ.functor.map P.π' ≫ eX.hom = (mapResolution F P).π' := |
| mapResolutionCochainIso_π F P |
| erw [hπ] at h |
| apply (SmallShiftedHom.precompEquiv (mapResolution F P).π' |
| (by rw [mem_quasiIso_iff]; infer_instance)).injective |
| change (SmallShiftedHom.mk₀ (quasiIso D (.up ℤ)) 0 rfl (mapResolution F P).π').comp |
| _ (add_zero (n : ℤ)) = |
| (SmallShiftedHom.mk₀ (quasiIso D (.up ℤ)) 0 rfl (mapResolution F P).π').comp |
| ((mapResolution F P).extEquivCohomologyClass.symm |
| (CohomologyClass.mk (mappedResolutionCocycle F P x))) (add_zero (n : ℤ)) |
| have hpost := precomp_extOfCocycle (mapResolution F P) (mappedResolutionCocycle F P x) |
| refine h.trans (Eq.trans ?_ hpost.symm) |
| apply congrArg (SmallShiftedHom.mk (quasiIso D (.up ℤ))) |
| rw [mappedResolutionCocycle, AddEquiv.symm_apply_apply] |
| simp only [ShiftedHom.mk₀_comp, ShiftedHom.comp_mk₀] |
| rfl |
|
|
| end ArExplicit.ResolutionProducts |
|
|
| end |
|
|
| end OAI |
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|
