The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 54.54.1. 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 54.32.4 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. $\square$


Comments (1)

Comment #3539 by Dario Weißmann on

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


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