instIsIsoFromLeftDerivedZero'Self

English

The map from opcycles 0 to F.obj X given by fromLeftDerivedZero' is an isomorphism.

Русский

Обратная картина от LeftDerivedZero' является изоморфизмом.

LaTeX

$$$\text{FromLeftDerivedZero'}: ((F.mapHomologicalComplex _).obj P.complex).opcycles 0 \to F(X) \text{ is an isomorphism}$$$

Lean4

instance (F : C ⥤ D) [F.Additive] (X : C) [Projective X] :
    IsIso ((ProjectiveResolution.self X).fromLeftDerivedZero' F) :=
  by
  dsimp [ProjectiveResolution.fromLeftDerivedZero']
  rw [ChainComplex.isIso_descOpcycles_iff]
  refine ⟨ShortComplex.Splitting.exact ?_, inferInstance⟩
  exact
    { r := 0
      s := 𝟙 _
      f_r := (F.map_isZero (isZero_zero _)).eq_of_src _ _ }

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