## 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}
```

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