preservesRightHomology_of_zero_g — Mathlib

English

If a short complex S has f = 0 and the cokernel of f is preserved by F, then F preserves the left homology of S.

Русский

Если короткий комплекс S имеет f = 0 и кoкernel f сохраняется F, тогда F сохраняет левую гомологию S.

LaTeX

$$$S.f = 0 \quad\Rightarrow\quad F.PreservesLeftHomologyOf S$$$

Lean4

/-- If a short complex `S` is such that `S.g = 0` and that the cokernel of `S.f` is preserved
by a functor `F`, then `F` preserves the right homology of `S`. -/
theorem preservesRightHomology_of_zero_g (hg : S.g = 0) [PreservesColimit (parallelPair S.f 0) F] :
    F.PreservesRightHomologyOf S :=
  ⟨fun h =>
    { f := by infer_instance
      g' := Limits.preservesKernel_zero' _ _ (by rw [← cancel_epi h.p, h.p_g', comp_zero, hg]) }⟩

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