The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 36.36.4. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Let $x_1, \ldots , x_ n \in X_ s$. Assume that

  1. $f$ is locally of finite type,

  2. $f$ is separated, and

  3. $x_1, \ldots , x_ n$ are pairwise distinct isolated points of $X_ s$.

Then there exists an elementary étale neighbourhood $(U, u) \to (S, s)$ and a decomposition

\[ U \times _ S X = W \amalg V_1 \amalg \ldots \amalg V_ n \]

into open and closed subschemes such that the morphisms $V_ i \to U$ are finite, the fibres of $V_ i \to U$ over $u$ are singletons $\{ v_ i\} $, each $v_ i$ maps to $x_ i$ with $\kappa (x_ i) = \kappa (v_ i)$, and the fibre of $W \to U$ over $u$ contains no points mapping to any of the $x_ i$.

Proof. Choose $(U, u) \to (S, s)$ and $V_ i \subset X_ U$ as in Lemma 36.36.2. Since $X_ U \to U$ is separated (Schemes, Lemma 25.21.13) and $V_ i \to U$ is finite hence proper (Morphisms, Lemma 28.42.11) we see that $V_ i \subset X_ U$ is closed by Morphisms, Lemma 28.39.7. Hence $V_ i \cap V_ j$ is a closed subset of $V_ i$ which does not contain $v_ i$. Hence the image of $V_ i \cap V_ j$ in $U$ is a closed set (because $V_ i \to U$ proper) not containing $u$. After shrinking $U$ we may therefore assume that $V_ i \cap V_ j = \emptyset $ for all $i, j$. This gives the decomposition as in the lemma. $\square$


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