# The Stacks Project

## Tag 011K

Lemma 12.13.11. Notation and assumptions as in Lemma 12.13.10 above. Assume in addition that $\mathcal{A}$ is abelian. The morphism of complexes $\delta : C^\bullet \to A[1]^\bullet$ induces the maps $$H^i(\delta) : H^i(C^\bullet) \longrightarrow H^i(A[1]^\bullet) = H^{i + 1}(A^\bullet)$$ which occur in the long exact homology sequence associated to the short exact sequence of cochain complexes by Lemma 12.12.12.

Proof. Omitted. $\square$

The code snippet corresponding to this tag is a part of the file homology.tex and is located in lines 3137–3151 (see updates for more information).

\begin{lemma}
\label{lemma-ses-termwise-split-long-cochain}
Notation and assumptions as in
Lemma \ref{lemma-ses-termwise-split-cochain} above.
Assume in addition that $\mathcal{A}$ is abelian.
The morphism of complexes $\delta : C^\bullet \to A[1]^\bullet$
induces the maps
$$H^i(\delta) : H^i(C^\bullet) \longrightarrow H^i(A[1]^\bullet) = H^{i + 1}(A^\bullet)$$
which occur in the long exact homology sequence associated
to the short exact sequence of cochain complexes by
Lemma \ref{lemma-long-exact-sequence-cochain}.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

Comment #2795 by Tim on September 3, 2017 a 5:13 pm UTC

Is there a reference, where the proof is carried out in detail?

Comment #2898 by Johan (site) on October 7, 2017 a 3:21 pm UTC

Why don't you try yourself?

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