The Stacks project

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

\[ Q = 0 \Leftrightarrow H^ i(X, Q) = 0 \text{ for all }i \]

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 $q$. $\square$

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