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
Comment #3671 by Johan on
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