Theorem 57.52.1. 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 57.51.6 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 57.33.1 and Limits, Lemma 32.2.1. 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 57.51.5. $\square$

Comment #2107 by Keenan Kidwell on

I'm confused by the strict Henselization appearing in the statement of the theorem. According to 57.33.1, $\mathscr{O}_{S,\overline{s}}$, the stalk of the étale 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.

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