| import OAI.Algebra.AuslanderReiten.Resolutions.ResolutionProductCompatibility |
|
|
| namespace OAI |
|
|
| 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] [HasDerivedCategory C] |
|
|
| def conjugateTo {X X' Y Y' : DerivedCategory C} (e : X ≅ X') (t : Y ≅ Y') |
| {n : ℤ} (f : ShiftedHom X Y n) : ShiftedHom X' Y' n := |
| e.inv ≫ f ≫ (CategoryTheory.shiftFunctor (DerivedCategory C) n).map t.hom |
|
|
| theorem conjugateTo_comp {X X' Y Y' Z Z' : DerivedCategory C} |
| (e : X ≅ X') (t : Y ≅ Y') (w : Z ≅ Z') |
| {n m r : ℤ} (f : ShiftedHom X Y n) (g : ShiftedHom Y Z m) (h : m+n=r) : |
| conjugateTo e w (f.comp g h) = (conjugateTo e t f).comp (conjugateTo t w g) h := by |
| simp only [conjugateTo, ShiftedHom.comp, Functor.map_comp, Category.assoc] |
| rw [← Category.assoc ((CategoryTheory.shiftFunctor (DerivedCategory C) n).map t.hom), |
| ← Functor.map_comp, t.hom_inv_id, CategoryTheory.Functor.map_id, Category.id_comp] |
| exact congrArg (fun z => e.inv ≫ f ≫ |
| (CategoryTheory.shiftFunctor (DerivedCategory C) n).map g ≫ z) |
| ((CategoryTheory.shiftFunctorAdd' (DerivedCategory C) m n r h).inv.naturality w.hom).symm |
|
|
| variable [HasExt.{v} C] |
|
|
| def extOfCocycle {X Y : C} (P : ProjectiveResolution X) {n : ℕ} |
| (y : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ)) : |
| Ext X Y n := P.extEquivCohomologyClass.symm (CohomologyClass.mk y) |
|
|
| theorem extOfCocycle_hom {X Y : C} (P : ProjectiveResolution X) {n : ℕ} |
| (y : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ)) : |
| (extOfCocycle P y).hom = conjugateTo (resolutionIso P) |
| ((DerivedCategory.singleFunctorIsoCompQ C 0).symm.app Y) |
| (ShiftedHom.map (Cocycle.equivHomShift.symm y) DerivedCategory.Q) := by |
| rw [extOfCocycle, P.extEquivCohomologyClass_symm_mk_hom] |
| simp only [conjugateTo, resolutionIso, Iso.trans_inv, asIso_inv, |
| ShiftedHom.mk₀_comp, ShiftedHom.comp_mk₀, Category.assoc] |
| rfl |
|
|
| theorem endExt_comp_extOfCocycle {X Y : C} (P : ProjectiveResolution X) {n m r : ℕ} |
| (x : Cocycle P.cochainComplex P.cochainComplex (n : ℤ)) |
| (y : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (m : ℤ)) |
| (z : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (r : ℤ)) |
| (h : n+m=r) (hz : z.val=x.val.comp y.val (by omega)) : |
| (endExt P x).comp (extOfCocycle P y) h = extOfCocycle P z := by |
| apply Ext.homEquiv.injective |
| change ((endExt P x).comp (extOfCocycle P y) h).hom = (extOfCocycle P z).hom |
| rw [Ext.comp_hom, endExt_hom, extOfCocycle_hom, extOfCocycle_hom, |
| cocycleHom_comp (by omega) x y z hz, ShiftedHom.map_comp] |
| exact (conjugateTo_comp (resolutionIso P) (resolutionIso P) |
| ((DerivedCategory.singleFunctorIsoCompQ C 0).symm.app Y) _ _ _).symm |
|
|
| end ArExplicit.ResolutionProducts |
|
|
| end |
|
|
| end OAI |
|
|
