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\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*}

Theorem 54.52.1. 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 54.51.4 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 54.33.1 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 54.51.3. $\square$


Comments (2)

Comment #2107 by Keenan Kidwell on

I'm confused by the strict Henselization appearing in the statement of the theorem. According to 54.33.1, , the stalk of the étale structure sheaf of at the geometric point , is already strictly Henselian right? So is the theorem really meant to have in place of , the latter being, as far as I can tell, the same thing as ? Of course this doesn't change the content of the theorem.

There are also:

  • 2 comment(s) on Section 54.52: Stalks of higher direct images

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