Lemma 13.27.9. Let $\mathcal{A}$ be an abelian category. Let $K$ be an object of $D^ b(\mathcal{A})$ such that $\mathop{\mathrm{Ext}}\nolimits ^ p_\mathcal {A}(H^ i(K), H^ j(K)) = 0$ for all $p \geq 2$ and $i > j$. Then $K$ is isomorphic to the direct sum of its cohomologies: $K \cong \bigoplus H^ i(K)[-i]$.

**Proof.**
Choose $a, b$ such that $H^ i(K) = 0$ for $i \not\in [a, b]$. We will prove the lemma by induction on $b - a$. If $b - a \leq 0$, then the result is clear. If $b - a > 0$, then we look at the distinguished triangle of truncations

see Remark 13.12.4. By Lemma 13.4.11 if the last arrow is zero, then $K \cong \tau _{\leq b - 1}K \oplus H^ b(K)[-b]$ and we win by induction. Again using induction we see that

By assumption the direct sum is zero and the proof is complete. $\square$

## 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Your email address will not be published.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)