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Typos in etale-cohomology
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aisejohan committed Oct 9, 2017
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8 changes: 4 additions & 4 deletions etale-cohomology.tex
Expand Up @@ -746,7 +746,7 @@ \section{Sheafification}
V \ar[rr]^\chi & & U.
}
$$
Given the data $\chi, \alpha, \{\chi_j\}_{i\in J}$ we define
Given the data $\chi, \alpha, \{\chi_j\}_{j \in J}$ we define
\begin{eqnarray*}
\check H^0(\mathcal{U}, \mathcal{F}) & \longrightarrow &
\check H^0(\mathcal{V}, \mathcal{F}) \\
Expand All @@ -757,7 +757,7 @@ \section{Sheafification}
\begin{enumerate}
\item the map is well-defined, and
\item depends only on $\chi$ and is independent of the choice of
$\alpha, \{\chi_j\}_{i\in J}$.
$\alpha, \{\chi_j\}_{j \in J}$.
\end{enumerate}
We omit the proof of the first fact.
To see part (2), consider another triple $(\psi, \beta, \psi_j)$ with
Expand All @@ -773,9 +773,9 @@ \section{Sheafification}
$$
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
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