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Additional lemma in limits

On properties when you glue in near a point...
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aisejohan committed Sep 20, 2018
1 parent f305cb8 commit bc7104e8ae0686eea66f4868c27da16708a75fa1
Showing with 29 additions and 4 deletions.
  1. +29 −4 limits.tex
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@@ -4503,6 +4503,28 @@ \section{Glueing in closed fibres}
of these results.
\end{proof}
\begin{lemma}
\label{lemma-glueing-near-point-properties}
Notation and assumptions as in Lemma \ref{lemma-glueing-near-point}.
Let $f' : X \to U'$ correspond to $f : X' \to U$ and
$g : Y \to \Spec(\mathcal{O}_{S, s})$
via the equivalence. If $f$ and $g$ are separated, proper, finite, \'etale,
then after possibly shrinking $U'$ the morphism $f'$ has the same property.
\end{lemma}
\begin{proof}
Recall that $\Spec(\mathcal{O}_{S, s})$ is the limit of the
affine open neighbourhoods of $s$ in $S$. Since $g$ has the property
in question, then the restriction of $f'$ to one of these
affine open neighbourhoods does too, see
Lemmas \ref{lemma-descend-separated-finite-presentation},
\ref{lemma-eventually-proper},
\ref{lemma-descend-finite-finite-presentation}, and
\ref{lemma-descend-etale}.
Since $f'$ has the given property over $U$ as $f$ does,
we conclude as one can check the property locally on the base.
\end{proof}
\begin{lemma}
\label{lemma-glueing-near-multiple-closed-points}
Let $S$ be a scheme. Let $s_1, \ldots, s_n \in S$ be pairwise distinct
@@ -4583,7 +4605,7 @@ \section{Application to modifications}
\label{lemma-modifications-properties}
Notation and assumptions as in Lemma \ref{lemma-modifications}.
Let $f : X \to S$ correspond to $g : Y \to \Spec(\mathcal{O}_{S, s})$
via the equivalence. Then $f$ is separated, proper, finite,
via the equivalence. Then $f$ is separated, proper, finite, \'etale
and add more here if and only if $g$ is so.
\end{lemma}
@@ -4595,10 +4617,13 @@ \section{Application to modifications}
Morphisms, Lemmas \ref{morphisms-lemma-base-change-proper} and
\ref{morphisms-lemma-base-change-finite}.
Hence if $f$ has the property, then so does $g$.
Conversely, if $g$ does, then $f$ does in a neighbourhood of $s$ by
The converse follows from Lemma \ref{lemma-glueing-near-point-properties}
but we also give a direct proof here.
Namely, if $g$ has to property, then $f$ does in a neighbourhood of $s$ by
Lemmas \ref{lemma-descend-separated-finite-presentation},
\ref{lemma-eventually-proper}, and
\ref{lemma-descend-finite-finite-presentation}.
\ref{lemma-eventually-proper},
\ref{lemma-descend-finite-finite-presentation}, and
\ref{lemma-descend-etale}.
Since $f$ clearly has the given property over $S \setminus \{s\}$
we conclude as one can check the property locally on the base.
\end{proof}

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