# stacks/stacks-project

On properties when you glue in near a point...
 @@ -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}