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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 | 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]
theorem cocycleHom_ofHom {K L : CochainComplex C ℤ} (f : K ⟶ L) :
Cocycle.equivHomShift.symm (Cocycle.ofHom f) = ShiftedHom.mk₀ 0 rfl f := by
apply HomologicalComplex.Hom.ext
funext i
simp only [CompleteComparison.cocycleHom_component, Cocycle.ofHom_coe]
simp only [ShiftedHom.mk₀, HomologicalComplex.comp_f, CategoryTheory.shiftFunctorZero',
eqToIso_refl, Iso.trans_inv, Iso.refl_inv,
CochainComplex.shiftFunctorZero_inv_app_f, Category.comp_id]
change (Cochain.ofHom f).v i (i+0) rfl =
f.f i ≫ (L.XIsoOfEq (show i=i+0 by omega)).hom
simpa only [Cochain.ofHom_v] using
(Cochain.v_comp_XIsoOfEq_hom (Cochain.ofHom f) i i (i+0) (add_zero i) (by omega)).symm
variable [HasDerivedCategory C] [HasExt.{v} C]
theorem endExt_identity {X : C} (P : ProjectiveResolution X) :
endExt P (Cocycle.ofHom (𝟙 P.cochainComplex)) = Ext.mk₀ (𝟙 X) := by
apply Ext.homEquiv.injective
change (endExt P (Cocycle.ofHom (𝟙 P.cochainComplex))).hom = (Ext.mk₀ (𝟙 X)).hom
rw [endExt_hom, cocycleHom_ofHom, Ext.mk₀_hom]
erw [ShiftedHom.map_mk₀]
simp [conjugate, ShiftedHom.mk₀, ← NatTrans.naturality]
theorem extMk_augmentation {X : C} (P : ProjectiveResolution X) :
P.extMk (Y := X) (P.π.f 0) 1 rfl P.complex_d_comp_π_f_zero = Ext.mk₀ (𝟙 X) := by
rw [← endExt_identity P, endExt_eq_augmented]
unfold ProjectiveResolution.extMk
apply congrArg P.extEquivCohomologyClass.symm
apply congrArg CohomologyClass.mk
apply Subtype.ext
apply (Cochain.toSingleEquiv (show (0 : ℤ)+0=0 by omega)).injective
change (Cochain.toSingleMk ((P.cochainComplexXIso 0 0 rfl).hom ≫ P.π.f 0)
(by omega : (0 : ℤ)+0=0)).v 0 0 (by omega) ≫
(HomologicalComplex.singleObjXSelf (.up ℤ) 0 X).hom =
((Cochain.ofHom (𝟙 P.cochainComplex)).comp (Cochain.ofHom P.π') (add_zero 0)).v 0 0
(by omega) ≫ (HomologicalComplex.singleObjXSelf (.up ℤ) 0 X).hom
rw [Cochain.id_comp, Cochain.ofHom_v, Cochain.toSingleMk_v, P.π'_f_zero]
dsimp only [ChainComplex.single₀_obj_zero]
exact congrArg (fun morphism =>
morphism ≫ (HomologicalComplex.singleObjXSelf (.up ℤ) 0 X).hom)
(Category.assoc (P.cochainComplexXIso 0 0 rfl).hom (P.π.f 0)
(HomologicalComplex.singleObjXSelf (.up ℤ) 0 X).inv)
end ArExplicit.ResolutionProducts
end
end OAI
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