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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 | 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
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