Lemma 59.80.5. Let $f : X \to Y$ be a morphism of schemes where $X$ is an integral normal scheme with separably closed function field. Then $R^ qf_*\underline{M} = 0$ for $q > 0$ and any abelian group $M$.

**Proof.**
Recall that $R^ qf_*\underline{M}$ is the sheaf associated to the presheaf $V \mapsto H^ q_{\acute{e}tale}(V \times _ Y X, M)$ on $Y_{\acute{e}tale}$, see Lemma 59.51.6. If $V$ is affine, then $V \times _ Y X \to X$ is separated and étale. Hence $V \times _ Y X = \coprod U_ i$ is a disjoint union of open subschemes $U_ i$ of $X$, see Lemma 59.80.4. By Lemma 59.80.1 we see that $H^ q_{\acute{e}tale}(U_ i, M)$ is equal to $H^ q_{Zar}(U_ i, M)$. This vanishes by Cohomology, Lemma 20.20.2.
$\square$

## 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Your email address will not be published.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)