The Stacks project

Lemma 21.47.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O})$. If two out of three of $K, L, M$ are perfect then the third is also perfect.

Proof. First proof: Combine Lemmas 21.47.4, 21.45.4, and 21.46.6. Second proof (sketch): Say $K$ and $L$ are perfect. Let $U$ be an object of $\mathcal{C}$. After replacing $U$ by the members of a covering we may assume that $K|_ U$ and $L|_ U$ are represented by strictly perfect complexes $\mathcal{K}^\bullet $ and $\mathcal{L}^\bullet $. After replacing $U$ by the members of a covering we may assume the map $K|_ U \to L|_ U$ is given by a map of complexes $\alpha : \mathcal{K}^\bullet \to \mathcal{L}^\bullet $, see Lemma 21.44.8. Then $M|_ U$ is isomorphic to the cone of $\alpha $ which is strictly perfect by Lemma 21.44.2. $\square$

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