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Tag 03Q9

Chapter 53: Étale Cohomology > Section 53.52: Stalks of higher direct images

Theorem 53.52.1. Let $f: X \to S$ be a quasi-compact and quasi-separated morphism of schemes, $\mathcal{F}$ an abelian sheaf on $X_{\acute{e}tale}$, and $\overline{s}$ a geometric point of $S$ lying over $s \in S$. Then $$ \left(R^nf_* \mathcal{F}\right)_{\overline{s}} = H_{\acute{e}tale}^n( X \times_S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^{sh}), p^{-1}\mathcal{F}) $$ where $p : X \times_S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^{sh}) \to X$ is the projection.

Proof. Let $\mathcal{I}$ be the category of étale neighborhoods of $\overline{s}$ on $S$. By Lemma 53.51.4 we have $$ (R^nf_*\mathcal{F})_{\overline{s}} = \mathop{\mathrm{colim}}\nolimits_{(V, \overline{v}) \in \mathcal{I}^{opp}} H_{\acute{e}tale}^n(X \times_S V, \mathcal{F}|_{X \times_S V}). $$ We may replace $\mathcal{I}$ by the initial subcategory consisting of affine étale neighbourhoods of $\overline{s}$. Observe that $$ \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^{sh}) = \mathop{\mathrm{lim}}\nolimits_{(V, \overline{v}) \in \mathcal{I}} V $$ by Lemma 53.33.1 and Limits, Lemma 31.2.1. Since fibre products commute with limits we also obtain $$ X \times_S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^{sh}) = \mathop{\mathrm{lim}}\nolimits_{(V, \overline{v}) \in \mathcal{I}} X \times_S V $$ We conclude by Lemma 53.51.3. $\square$

    The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 7371–7383 (see updates for more information).

    \begin{theorem}
    \label{theorem-higher-direct-images}
    Let $f: X \to S$ be a quasi-compact and quasi-separated morphism of schemes,
    $\mathcal{F}$ an abelian sheaf on $X_\etale$, and $\overline{s}$ a
    geometric point of $S$ lying over $s \in S$. Then
    $$
    \left(R^nf_* \mathcal{F}\right)_{\overline{s}} =
    H_\etale^n( X \times_S \Spec(\mathcal{O}_{S, s}^{sh}),
    p^{-1}\mathcal{F})
    $$
    where $p : X \times_S \Spec(\mathcal{O}_{S, s}^{sh}) \to X$
    is the projection.
    \end{theorem}
    
    \begin{proof}
    Let $\mathcal{I}$ be the category of \'etale neighborhoods of $\overline{s}$
    on $S$. By Lemma \ref{lemma-higher-direct-images}
    we have
    $$
    (R^nf_*\mathcal{F})_{\overline{s}} =
    \colim_{(V, \overline{v}) \in \mathcal{I}^{opp}}
    H_\etale^n(X \times_S V, \mathcal{F}|_{X \times_S V}).
    $$
    We may replace $\mathcal{I}$ by the initial subcategory consisting
    of affine \'etale neighbourhoods of $\overline{s}$. Observe that
    $$
    \Spec(\mathcal{O}_{S, s}^{sh}) =
    \lim_{(V, \overline{v}) \in \mathcal{I}} V
    $$
    by Lemma \ref{lemma-describe-etale-local-ring} and
    Limits, Lemma
    \ref{limits-lemma-directed-inverse-system-affine-schemes-has-limit}.
    Since fibre products commute with limits we also obtain
    $$
    X \times_S \Spec(\mathcal{O}_{S, s}^{sh}) =
    \lim_{(V, \overline{v}) \in \mathcal{I}} X \times_S V
    $$
    We conclude by Lemma \ref{lemma-directed-colimit-cohomology}.
    \end{proof}

    Comments (2)

    Comment #2107 by Keenan Kidwell on July 11, 2016 a 6:26 pm UTC

    I'm confused by the strict Henselization appearing in the statement of the theorem. According to 04HX, $\mathscr{O}_{S,\overline{s}}$, the stalk of the ├ętale structure sheaf of $S$ at the geometric point $\overline{s}$, is already strictly Henselian right? So is the theorem really meant to have $\mathscr{O}_{S,s}^{\mathrm{sh}}$ in place of $\mathscr{O}_{S,\overline{s}}^{\mathrm{sh}}$, the latter being, as far as I can tell, the same thing as $\mathscr{O}_{S,\overline{s}}$? Of course this doesn't change the content of the theorem.

    Comment #2133 by Johan (site) on July 21, 2016 a 3:26 pm UTC

    Yep, you are right. See change here.

    There are also 2 comments on Section 53.52: Étale Cohomology.

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