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)

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