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Fix indices in limits

Thanks to Minseon Shin
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aisejohan committed Nov 15, 2019
1 parent debce27 commit 21512aed9cf27f598bfa08eeb346b3e14ff70320
Showing with 7 additions and 6 deletions.
  1. +7 −6 limits.tex
@@ -1061,8 +1061,8 @@ \section{Absolute Noetherian Approximation}
\begin{enumerate}
\item the transition morphisms $g_{jj'} : S_j \to S_{j'}$ are affine,
\item each $S_j$ is of finite type over $\mathbf{Z}$,
\item $g_{jj'}^{-1}(V_{j'}) = V_j$,
\item $S = \lim S_j$ and $V = \lim V_j$, and
\item $g_{jj'}^{-1}(V'_{j'}) = V'_j$,
\item $S = \lim S_j$ and $V = \lim V'_j$, and
\item the diagrams
$$
\vcenter{
@@ -1074,8 +1074,8 @@ \section{Absolute Noetherian Approximation}
\quad\text{and}\quad
\vcenter{
\xymatrix{
V_j \ar[r] \ar[d] & V_{\alpha(j)} \ar[d] \\
V_{j'} \ar[r] & V_{\alpha(j')}
V'_j \ar[r] \ar[d] & V_{\alpha(j)} \ar[d] \\
V'_{j'} \ar[r] & V_{\alpha(j')}
}
}
$$
@@ -1106,10 +1106,11 @@ \section{Absolute Noetherian Approximation}
an inverse system over a directed set $I$, each $f_{ii'}$
affine and each $V_i$ of finite type over $\mathbf{Z}$.
\end{enumerate}
Denote $f_i : V \to V_i$ the projections.
Set $W = U \cap V$. As $S$ is quasi-separated, this is a quasi-compact open
of $V$. By Lemma \ref{lemma-descend-opens}
(and after shrinking $I$) we may assume that there exist
opens $W_i \subset V_i$ such that $f_{ij}^{-1}(W_j) = W_i$
opens $W_i \subset V_i$ such that $f_{ii'}^{-1}(W_{i'}) = W_i$
and such that $f_i^{-1}(W_i) = W$. Since $W$ is a quasi-compact open
of $U$ it is quasi-affine. Hence we may assume (after shrinking $I$ again)
that $W_i$ is quasi-affine for all $i$, see
@@ -1165,7 +1166,7 @@ \section{Absolute Noetherian Approximation}
assume the corresponding opens $D(g_{l, i}) \subset \Spec(R_i)$
are contained in $W_i$ for $l = 1, \ldots, m$ and cover $W_i$.
We conclude that the morphism $W_i \to \Spec(R_i) \to \Spec(B_i)$
is an open immersion, see Lemma \ref{lemma-diagram}
is an open immersion, see Lemma \ref{lemma-diagram}.

\medskip\noindent
By Lemma \ref{lemma-quasi-affine-finite-type-over-Z}

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