The Stacks project

Lemma 59.55.1. Let $R$ be a strictly henselian local ring. Set $S = \mathop{\mathrm{Spec}}(R)$ and let $\overline{s}$ be its closed point. Then the global sections functor $\Gamma (S, -) : \textit{Ab}(S_{\acute{e}tale}) \to \textit{Ab}$ is exact. In fact we have $\Gamma (S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$ for any sheaf of sets $\mathcal{F}$. In particular

\[ \forall p\geq 1, \quad H_{\acute{e}tale}^ p(S, \mathcal{F})=0 \]

for all $\mathcal{F}\in \textit{Ab}(S_{\acute{e}tale})$.

Proof. If we show that $\Gamma (S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$ then $\Gamma (S, -)$ is exact as the stalk functor is exact. Let $(U, \overline{u})$ be an étale neighbourhood of $\overline{s}$. Pick an affine open neighborhood $\mathop{\mathrm{Spec}}(A)$ of $\overline{u}$ in $U$. Then $R \to A$ is étale and $\kappa (\overline{s}) = \kappa (\overline{u})$. By Theorem 59.32.4 we see that $A \cong R \times A'$ as an $R$-algebra compatible with maps to $\kappa (\overline{s}) = \kappa (\overline{u})$. Hence we get a section

\[ \xymatrix{ \mathop{\mathrm{Spec}}(A) \ar[r] & U \ar[d]\\ & S \ar[ul] } \]

It follows that in the system of étale neighbourhoods of $\overline{s}$ the identity map $(S, \overline{s}) \to (S, \overline{s})$ is cofinal. Hence $\Gamma (S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$. The final statement of the lemma follows as the higher derived functors of an exact functor are zero, see Derived Categories, Lemma 13.16.9. $\square$


Comments (2)

Comment #3539 by Dario Weißmann on

Typo in the first line of the proof: ... the is exact ...

There are also:

  • 1 comment(s) on Section 59.55: Vanishing of finite higher direct images

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Your email address will not be published.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 03QO. Beware of the difference between the letter 'O' and the digit '0'.