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 54.89.3. Let $K$ be a separably closed field. Let $X$ be a scheme of finite type over $K$. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$ whose support is contained in the set of closed points of $X$. Then $H^ q(X, \mathcal{F}) = 0$ for $q > 0$ and $\mathcal{F}$ is globally generated.

Proof. (If $\mathcal{F}$ is torsion, then the vanishing follows immediately from Lemma 54.87.7.) By Lemma 54.73.5 we can write $\mathcal{F}$ as a filtered colimit of constructible sheaves $\mathcal{F}_ i$ of $\mathbf{Z}$-modules whose supports $Z_ i \subset X$ are finite sets of closed points. By Proposition 54.46.4 such a sheaf is of the form $(Z_ i \to X)_*\mathcal{G}_ i$ where $\mathcal{G}_ i$ is a sheaf on $Z_ i$. As $K$ is separably closed, the scheme $Z_ i$ is a finite disjoint union of spectra of separably closed fields. Recall that $H^ q(Z_ i, \mathcal{G}_ i) = H^ q(X, \mathcal{F}_ i)$ by the Leray spectral sequence for $Z_ i \to X$ and vanising of higher direct images for this morphism (Proposition 54.54.2). By Lemmas 54.58.1 and 54.58.2 we see that $H^ q(Z_ i, \mathcal{G}_ i)$ is zero for $q > 0$ and that $H^0(Z_ i, \mathcal{G}_ i)$ generates $\mathcal{G}_ i$. We conclude the vanishing of $H^ q(X, \mathcal{F}_ i)$ for $q > 0$ and that $\mathcal{F}_ i$ is generated by global sections. By Theorem 54.51.3 we see that $H^ q(X, \mathcal{F}) = 0$ for $q > 0$. The proof is now done because a filtered colimit of globally generated sheaves of abelian groups is globally generated (details omitted). $\square$


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