Lemma 21.26.2. In the situation above, let $K_1 \to K_2 \to K_3 \to K_1[1]$ be a distinguished triangle in $D(\mathcal{O})$. If $c^{K_ i}_{X, Z, Y, E}$ is a quasi-isomorphism for two $i$ out of $\{ 1, 2, 3\} $, then it is a quasi-isomorphism for the third $i$.

**Proof.**
By rotating the triangle we may assume $c^{K_1}_{X, Z, Y, E}$ and $c^{K_2}_{X, Z, Y, E}$ are quasi-isomorphisms. Choose a map $f : \mathcal{I}^\bullet _1 \to \mathcal{I}^\bullet _2$ of K-injective complexes of $\mathcal{O}$-modules representing $K_1 \to K_2$. Then $K_3$ is represented by the K-injective complex $C(f)^\bullet $, see Derived Categories, Lemma 13.31.3. Then the morphism $c^{K_3}_{X, Z, Y, E}$ is an isomorphism as it is the third leg in a map of distinguished triangles in $K(\textit{Ab})$ whose other two legs are quasi-isomorphisms. Some details omitted; use Derived Categories, Lemma 13.4.3.
$\square$

## 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Your email address will not be published.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.

## Comments (0)