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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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