descFZero — Mathlib

English

In a short exact sequence S, HasInjectiveDimensionLT X3 (n+1) is equivalent to HasInjectiveDimensionLT X1 (n+2) provided X2 is Injective.

Русский

В SE имеет LT X3 (n+1) тогда и только тогда LT X1 (n+2), при условии, что X2 injective.

LaTeX

$$$S.ShortExact \land \text{Injective}(S.X_2) \Rightarrow (\text{HasInjectiveDimensionLT}(S.X_3,n+1) \iff \text{HasInjectiveDimensionLT}(S.X_1,n+2))$$$

Lean4

/-- Auxiliary construction for `desc`. -/
def descFZero {Y Z : C} (f : Z ⟶ Y) (I : InjectiveResolution Y) (J : InjectiveResolution Z) :
    J.cocomplex.X 0 ⟶ I.cocomplex.X 0 :=
  factorThru (f ≫ I.ι.f 0) (J.ι.f 0)

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