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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | 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
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