| import OAI.Algebra.AuslanderReiten.Resolutions.ResolutionHomology |
| import OAI.Algebra.AuslanderReiten.Resolutions.ResolutionExtIdentity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits |
| namespace ArExplicit.ResolutionHomology |
| universe u v |
| variable {C : Type u} [Category.{v} C] [Abelian C] [HasExt.{v} C] |
| variable {X Y : C} (P : ProjectiveResolution X) |
|
|
| theorem extMk_zeroDegree_eq_zero_iff (f : P.complex.X 0 ⟶ Y) |
| (hf : P.complex.d 1 0 ≫ f = 0) : P.extMk f 1 rfl hf = 0 ↔ f = 0 := by |
| let := HasDerivedCategory.standard C |
| obtain ⟨g,hg⟩ := CokernelCofork.IsColimit.desc' P.isColimitCokernelCofork f hf |
| change X ⟶ Y at g |
| change P.π.f 0 ≫ g = f at hg |
| have he : P.extMk (P.π.f 0 ≫ g) 1 rfl |
| (by erw [← Category.assoc, P.complex_d_comp_π_f_zero, zero_comp]) = Ext.mk₀ g := by |
| erw [← P.extMk_comp_mk₀ (P.π.f 0) 1 rfl P.complex_d_comp_π_f_zero g, |
| ResolutionProducts.extMk_augmentation, Ext.mk₀_id_comp] |
| rfl |
| have he' : P.extMk f 1 rfl hf = Ext.mk₀ g := by simpa only [hg] using he |
| constructor |
| · intro hz |
| have hg0 : g=0 := Ext.homEquiv₀.symm.injective (by |
| change Ext.mk₀ g = Ext.mk₀ (0 : X ⟶ Y) |
| erw [← he', hz, Ext.mk₀_zero]) |
| erw [← hg, hg0, comp_zero] |
| · intro hz |
| cases hz |
| exact P.extMk_zero 1 rfl |
|
|
| end ArExplicit.ResolutionHomology |
|
|
| end |
|
|
| end OAI |
|
|
