The Stacks project

Lemma 59.97.3. 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 59.95.7.) By Lemma 59.74.5 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 59.46.4 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 59.55.2). By Lemmas 59.59.1 and 59.59.2 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 59.51.3 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). $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F1F. Beware of the difference between the letter 'O' and the digit '0'.