# stacks/stacks-project

Change an index

Thanks to Yu-Liang Huang
http://stacks.math.columbia.edu/tag/03Q7#comment-2356
 @@ -7414,8 +7414,8 @@ \section{Stalks of higher direct images} $\mathcal{F}$ an abelian sheaf on $X_\etale$, and $\overline{s}$ a geometric point of $S$ lying over $s \in S$. Then $$-\left(R^pf_* \mathcal{F}\right)_{\overline{s}} = -H_\etale^p( X \times_S \Spec(\mathcal{O}_{S, s}^{sh}), +\left(R^nf_* \mathcal{F}\right)_{\overline{s}} = +H_\etale^n( X \times_S \Spec(\mathcal{O}_{S, s}^{sh}), p^{-1}\mathcal{F})$$ where $p : X \times_S \Spec(\mathcal{O}_{S, s}^{sh}) \to X$ @@ -7427,9 +7427,9 @@ \section{Stalks of higher direct images} on $S$. By Lemma \ref{lemma-higher-direct-images} we have $$-(R^pf_*\mathcal{F})_{\overline{s}} = +(R^nf_*\mathcal{F})_{\overline{s}} = \colim_{(V, \overline{v}) \in \mathcal{I}^{opp}} -H^p(X \times_S V, \mathcal{F}|_{X \times_S V}). +H^n(X \times_S V, \mathcal{F}|_{X \times_S V}).$$ We may replace $\mathcal{I}$ by the initial subcategory consisting of affine \'etale neighbourhoods of $\overline{s}$. Observe that