Lemma 12.14.4. Let $\mathcal{A}$ be an abelian category. Let

\[ 0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 \]

be a short exact sequence of complexes. Suppose that $\{ s_ n : C_ n \to B_ n\} $ is a family of morphisms which split the short exact sequences $0 \to A_ n \to B_ n \to C_ n \to 0$. Let $\pi _ n : B_ n \to A_ n$ be the associated projections, see Lemma 12.5.10. Then the family of morphisms

\[ \pi _{n - 1} \circ d_{B, n} \circ s_ n : C_ n \to A_{n - 1} \]

define a morphism of complexes $\delta (s) : C_\bullet \to A[-1]_\bullet $.

**Proof.**
Denote $i : A_\bullet \to B_\bullet $ and $q : B_\bullet \to C_\bullet $ the maps of complexes in the short exact sequence. Then $i_{n - 1} \circ \pi _{n - 1} \circ d_{B, n} \circ s_ n = d_{B, n} \circ s_ n - s_{n - 1} \circ d_{C, n}$. Hence $i_{n - 2} \circ d_{A, n - 1} \circ \pi _{n - 1} \circ d_{B, n} \circ s_ n = d_{B, n - 1} \circ (d_{B, n} \circ s_ n - s_{n - 1} \circ d_{C, n}) = - d_{B, n - 1} \circ s_{n - 1} \circ d_{C, n}$ as desired.
$\square$

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