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editors_note.tex
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\chapter*{Editors' notes}
We have tried to keep the book as similar to the original with minor changes.
Here are some changes in notation:
\begin{longtable}{|c|c|p{9cm}|}
\hline
\textbf{Original} & \textbf{New} & \textbf{Meaning} \\
\hline
\endhead
\hline
\endfoot
$\Sigma_K$ & $M_K^0$ & Set of finite places of a number field $K$. \\
% $D_w$ & $G_w$ & The decomposition group of a field extension $L/K$ relative to a place $w$ of $L$. \\
$\ell$ & $\lambda$ & The residue field of a field $L$ relative to a finite place. \\
$R^*$ & $R^\times$ & The group of units of a ring $R$. \\
$U^\circ$ & $\Int U$ & The interior of a subset $U$ of a topological space. \\
$A_K$ & $\mathcal{O}_K$ & The ring of algebraic integers of a number field $K$. \\
$\operatorname{N} v$ & $\numnorm v$ & ${} = [\mathcal{O}_v : \mathfrak{m}_v].$ \\
$\GG_{m/K}$ & $\GG_{m, K}$ & The multiplicative group of $K$. \\
$\PP_{n/K}$ & $\PP^n_K$ & The $n$-dimensional projective space over a field $K$. \\
$X \times_K L$ & $X \otimes_K L$ & The base change of a $K$-scheme $X$ by a field extension $L/K$. \\
\end{longtable}
We also did some minor corrections and errata we found:
\begin{itemize}
\item Page~\pageref{errata:t't} (I-3):
it originally said \textquote{$T'/T$}, and it should be
\textquote{$T/T'$}.
\item Page~\pageref{errata:An+k} (III-34):
it originally said \textquote{$A_n/A_{n+k}$}, and it should be
\textquote{$A_n/A_{n+1}$}.
\item Page~\pageref{errata:uv12} (IV-8):
it originally said \textquote{$\Delta_v = u^{12} \Delta'$}, and it
should be \textquote{$\Delta_v = u_v^{12} \Delta'$}.
\item Page~\pageref{errata:wrongsec} (IV-17):
it originally said \textquote{by the theorem on section~4}, but the
correct reference was \ref{sec:IV_22}.
\end{itemize}