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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08G9. Beware of the difference between the letter 'O' and the digit '0'.