The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 54.89.4. 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 54.89.3 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 54.89.3 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 54.89.3 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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