Lemma 59.97.4. Let $K$ be a separably closed field. Let $X$ be a scheme of finite type over $K$. Let $Q \in D(X_{\acute{e}tale})$. Assume that $Q_{\overline{x}}$ is nonzero only if $x$ is a closed point of $X$. Then

**Proof.**
The implication from left to right is trivial. Thus we need to prove the reverse implication.

Assume $Q$ is bounded below; this cases suffices for almost all applications. If $Q$ is not zero, then we can look at the smallest $i$ such that the cohomology sheaf $H^ i(Q)$ is nonzero. By Lemma 59.97.3 we have $H^ i(X, Q) = H^0(X, H^ i(Q)) \not= 0$ and we conclude.

General case. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ be the quasi-compact objects. By Lemma 59.97.3 the assumptions of Cohomology on Sites, Lemma 21.23.11 are satisfied. We conclude that $H^ q(U, Q) = H^0(U, H^ q(Q))$ for all $U \in \mathcal{B}$. In particular, this holds for $U = X$. Thus the conclusion by Lemma 59.97.3 as $Q$ is zero in $D(X_{\acute{e}tale})$ if and only if $H^ q(Q)$ is zero for all $q$. $\square$

## 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Your email address will not be published.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)