The stalks of higher direct images can often be computed as follows.
Theorem 59.53.1. 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
\[ \left(R^ nf_* \mathcal{F}\right)_{\overline{s}} = H_{\acute{e}tale}^ n( X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}\mathcal{F}) \]
where $p : X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) \to X$ is the projection. For $K \in D^+(X_{\acute{e}tale})$ and $n \in \mathbf{Z}$ we have
\[ \left(R^ nf_*K\right)_{\overline{s}} = H_{\acute{e}tale}^ n(X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K) \]
In fact, we have
\[ \left(Rf_*K\right)_{\overline{s}} = R\Gamma _{\acute{e}tale}(X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K) \]
in $D^+(\textit{Ab})$.
Proof.
Let $\mathcal{I}$ be the category of étale neighborhoods of $\overline{s}$ on $S$. By Lemma 59.51.6 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
\[ \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) = \mathop{\mathrm{lim}}\nolimits _{(V, \overline{v}) \in \mathcal{I}} V \]
by Lemma 59.33.1 and Limits, Lemma 32.2.1. Since fibre products commute with limits we also obtain
\[ X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) = \mathop{\mathrm{lim}}\nolimits _{(V, \overline{v}) \in \mathcal{I}} X \times _ S V \]
We conclude by Lemma 59.51.5. For the second variant, use the same argument using Lemma 59.52.3 instead of Lemma 59.51.5.
To see that the last statement is true, it suffices to produce a map $\left(Rf_*K\right)_{\overline{s}} \to R\Gamma _{\acute{e}tale}(X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K)$ in $D^+(\textit{Ab})$ which realizes the ismorphisms on cohomology groups in degree $n$ above for all $n$. To do this, choose a bounded below complex $\mathcal{J}^\bullet $ of injective abelian sheaves on $X_{\acute{e}tale}$ representing $K$. The complex $f_*\mathcal{J}^\bullet $ represents $Rf_*K$. Thus the complex
\[ (f_*\mathcal{J}^\bullet )_{\overline{s}} = \mathop{\mathrm{colim}}\nolimits _{(V, \overline{v}) \in \mathcal{I}^{opp}} (f_*\mathcal{J}^\bullet )(V) \]
represents $(Rf_*K)_{\overline{s}}$. For each $V$ we have maps
\[ (f_*\mathcal{J}^\bullet )(V) = \Gamma (X \times _ S V, \mathcal{J}^\bullet ) \longrightarrow \Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}\mathcal{J}^\bullet ) \]
and the target complex represents $R\Gamma _{\acute{e}tale}(X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}), p^{-1}K)$ in $D^+(\textit{Ab})$. Taking the colimit of these maps we obtain the result.
$\square$
Comments (2)
Comment #2356 by Yu-Liang Huang on
Comment #2423 by Johan on