The Stacks project

Lemma 13.9.16. Let $\mathcal{A}$ be an additive category. Let $(\alpha : A^\bullet \to B^\bullet , \beta : B^\bullet \to C^\bullet , s^ n, \pi ^ n)$ be a termwise split sequence of complexes. Let $(A^\bullet , B^\bullet , C^\bullet , \alpha , \beta , \delta )$ be the associated triangle. Then the triangle $(C^\bullet [-1], A^\bullet , B^\bullet , \delta [-1], \alpha , \beta )$ is isomorphic to the triangle $(C^\bullet [-1], A^\bullet , C(\delta [-1])^\bullet , \delta [-1], i, p)$.

Proof. We write $B^ n = A^ n \oplus C^ n$ and we identify $\alpha ^ n$ and $\beta ^ n$ with the natural inclusion and projection maps. By construction of $\delta $ we have

\[ d_ B^ n = \left( \begin{matrix} d_ A^ n & \delta ^ n \\ 0 & d_ C^ n \end{matrix} \right) \]

On the other hand the cone of $\delta [-1] : C^\bullet [-1] \to A^\bullet $ is given as $C(\delta [-1])^ n = A^ n \oplus C^ n$ with differential identical with the matrix above! Whence the lemma. $\square$

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