## Tag `03QO`

Chapter 53: Étale Cohomology > Section 53.54: Vanishing of finite higher direct images

Lemma 53.54.1. 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 $\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$ the $\Gamma(S, -)$ 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 53.32.4 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 13.17.9. $\square$

The code snippet corresponding to this tag is a part of the file `etale-cohomology.tex` and is located in lines 7479–7491 (see updates for more information).

```
\begin{lemma}
\label{lemma-vanishing-etale-cohomology-strictly-henselian}
Let $R$ be a strictly henselian local ring. Set $S = \Spec(R)$ and let
$\overline{s}$ be its closed point. Then the global
sections functor
$\Gamma(S, -) : \textit{Ab}(S_\etale) \to \textit{Ab}$
is exact. In fact we have $\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$
for any sheaf of sets $\mathcal{F}$. In particular
$$
\forall p\geq 1, \quad H_\etale^p(S, \mathcal{F})=0
$$
for all $\mathcal{F}\in \textit{Ab}(S_\etale)$.
\end{lemma}
\begin{proof}
If we show that $\Gamma(S, \mathcal{F}) = \mathcal{F}_{\overline{s}}$
the $\Gamma(S, -)$ is exact as the stalk functor is exact.
Let $(U, \overline{u})$ be an \'etale neighbourhood of $\overline{s}$.
Pick an affine open neighborhood $\Spec(A)$ of $\overline{u}$ in $U$.
Then $R \to A$ is \'etale and $\kappa(\overline{s}) = \kappa(\overline{u})$.
By Theorem \ref{theorem-henselian} 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
$$
\xymatrix{
\Spec(A) \ar[r] & U \ar[d]\\
& S \ar[ul]
}
$$
It follows that in the system of \'etale 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 \ref{derived-lemma-right-derived-exact-functor}.
\end{proof}
```

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