Lemma 57.78.1. Let $S$ be a scheme all of whose local rings are strictly henselian. Then for any abelian sheaf $\mathcal{F}$ on $S_{\acute{e}tale}$ we have $H^ i(S_{\acute{e}tale}, \mathcal{F}) = H^ i(S_{Zar}, \mathcal{F})$.

**Proof.**
Let $\epsilon : S_{\acute{e}tale}\to S_{Zar}$ be the morphism of sites given by the inclusion functor. The Zariski sheaf $R^ p\epsilon _*\mathcal{F}$ is the sheaf associated to the presheaf $U \mapsto H^ p_{\acute{e}tale}(U, \mathcal{F})$. Thus the stalk at $x \in X$ is $\mathop{\mathrm{colim}}\nolimits H^ p_{\acute{e}tale}(U, \mathcal{F}) = H^ p_{\acute{e}tale}(\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}), \mathcal{G}_ x)$ where $\mathcal{G}_ x$ denotes the pullback of $\mathcal{F}$ to $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$, see Lemma 57.51.5. Thus the higher direct images of $R^ p\epsilon _*\mathcal{F}$ are zero by Lemma 57.54.1 and we conclude by the Leray spectral sequence.
$\square$

## 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Your email address will not be published.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)