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1 parent 103d242 commit 5349282c6bf6e8fcedac1a4b9c04f838cd44aab7 @aisejohan aisejohan committed Feb 17, 2017
Showing with 4 additions and 4 deletions.
  1. +4 −4 etale-cohomology.tex
@@ -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

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