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 36.36.3. Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots , x_ n \in X$ be points having the same image $s$ in $S$. Assume that

  1. $f$ is locally of finite type, and

  2. $x_ i \in X_ s$ is isolated for $i = 1, \ldots , n$.

Then there exist

  1. an étale neighbourhood $(U, u) \to (S, s)$,

  2. for each $i$ an integer $m_ i$ and open subschemes $V_{i, j} \subset X_ U$, $j = 1, \ldots , m_ i$

such that we have

  1. each $V_{i, j} \to U$ is a finite morphism,

  2. there is a unique point $v_{i, j}$ of $V_{i, j}$ mapping to $u$ in $U$ with $\kappa (u) \subset \kappa (v_{i, j})$ finite purely inseparable,

  3. if $v_{i, j} = v_{i', j'}$, then $i = i'$ and $j = j'$, and

  4. the points $v_{i, j}$ map to $x_ i$ in $X$ and no other points of $(X_ U)_ u$ map to $x_ i$.

Proof. This proof is a variant of the proof of Algebra, Lemma 10.141.23 in the language of schemes. By Morphisms, Lemma 28.19.6 the morphism $f$ is quasi-finite at each of the points $x_ i$. Hence $\kappa (s) \subset \kappa (x_ i)$ is finite for each $i$ (Morphisms, Lemma 28.19.5). For each $i$, let $\kappa (s) \subset L_ i \subset \kappa (x_ i)$ be the subfield such that $L_ i/\kappa (s)$ is separable, and $\kappa (x_ i)/L_ i$ is purely inseparable. Choose a finite Galois extension $\kappa (s) \subset L$ such that there exist $\kappa (s)$-embeddings $L_ i \to L$ for $i = 1, \ldots , n$. Choose an étale neighbourhood $(U, u) \to (S, s)$ such that $L \cong \kappa (u)$ as $\kappa (s)$-extensions (Lemma 36.31.2).

Let $y_{i, j}$, $j = 1, \ldots , m_ i$ be the points of $X_ U$ lying over $x_ i \in X$ and $u \in U$. By Schemes, Lemma 25.17.5 these points $y_{i, j}$ correspond exactly to the primes in the rings $\kappa (u) \otimes _{\kappa (s)} \kappa (x_ i)$. This also explains why there are finitely many; in fact $m_ i = [L_ i : \kappa (s)]$ but we do not need this. By our choice of $L$ (and elementary field theory) we see that $\kappa (u) \subset \kappa (y_{i, j})$ is finite purely inseparable for each pair $i, j$. Also, by Morphisms, Lemma 28.19.13 for example, the morphism $X_ U \to U$ is quasi-finite at the points $y_{i, j}$ for all $i, j$.

Apply Lemma 36.36.2 to the morphism $X_ U \to U$, the point $u \in U$ and the points $y_{i, j} \in (X_ U)_ u$. This gives an étale neighbourhood $(U', u') \to (U, u)$ with $\kappa (u) = \kappa (u')$ and opens $V_{i, j} \subset X_{U'}$ with the properties (i), (ii), and (iii) of that lemma. We claim that the étale neighbourhood $(U', u') \to (S, s)$ and the opens $V_{i, j} \subset X_{U'}$ are a solution to the problem posed by the lemma. We omit the verifications. $\square$


Comments (2)

Comment #2993 by Dan Dore on

The numbering in the Lemma says "(\romannumeral1)", etc. instead of "(i)", etc.

Comment #3116 by on

THis is a problem with the website and not with the latex.

There are also:

  • 2 comment(s) on Section 36.36: Étale localization of quasi-finite morphisms

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