From 9346ae17958ab45041931eebeffb6ed46120b990 Mon Sep 17 00:00:00 2001 From: Aise Johan de Jong Date: Mon, 9 Oct 2017 08:57:52 -0400 Subject: [PATCH] Typos in etale-cohomology Thanks to Dario Weissmann https://stacks.math.columbia.edu/tag/03NQ#comment-2811 --- etale-cohomology.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/etale-cohomology.tex b/etale-cohomology.tex index 4a9922ace..dac856ef4 100644 --- a/etale-cohomology.tex +++ b/etale-cohomology.tex @@ -746,7 +746,7 @@ \section{Sheafification} V \ar[rr]^\chi & & U. } $$ -Given the data $\chi, \alpha, \{\chi_j\}_{i\in J}$ we define +Given the data $\chi, \alpha, \{\chi_j\}_{j \in J}$ we define \begin{eqnarray*} \check H^0(\mathcal{U}, \mathcal{F}) & \longrightarrow & \check H^0(\mathcal{V}, \mathcal{F}) \\ @@ -757,7 +757,7 @@ \section{Sheafification} \begin{enumerate} \item the map is well-defined, and \item depends only on $\chi$ and is independent of the choice of -$\alpha, \{\chi_j\}_{i\in J}$. +$\alpha, \{\chi_j\}_{j \in J}$. \end{enumerate} We omit the proof of the first fact. To see part (2), consider another triple $(\psi, \beta, \psi_j)$ with @@ -773,9 +773,9 @@ \section{Sheafification} $$ Given a section $s \in \mathcal{F}(\mathcal{U})$, its image in $\mathcal{F}(V_j)$ under the map given by -$(\chi, \alpha, \{\chi_j\}_{i\in J})$ +$(\chi, \alpha, \{\chi_j\}_{j \in J})$ is $\chi_j^*s_{\alpha(j)}$, and -its image under the map given by $(\psi, \beta, \{\psi_j\}_{i\in J})$ +its image under the map given by $(\psi, \beta, \{\psi_j\}_{j \in J})$ is $\psi_j^*s_{\beta(j)}$. These two are equal since by assumption $s \in \check H(\mathcal{U}, \mathcal{F})$ and hence both are equal to the pullback of the common value