The Stacks project

Lemma 37.46.1. Let $f : X \to S$ be a morphism. Let $x \in X$ and set $s = f(x)$. Assume that $f$ is locally of finite type and that $n = \dim _ x(X_ s)$. Then there exists a commutative diagram

\[ \xymatrix{ X \ar[dd] & X' \ar[l]^ g \ar[d]^\pi & x \ar@{|->}[dd] & x' \ar@{|->}[l] \ar@{|->}[d] \\ & Y \ar[d]^ h & & y \ar@{|->}[d] \\ S \ar@{=}[r] & S & s & s \ar@{=}[l] } \]

and a point $x' \in X'$ with $g(x') = x$ such that with $y = \pi (x')$ we have

  1. $h : Y \to S$ is smooth of relative dimension $n$,

  2. $g : (X', x') \to (X, x)$ is an elementary étale neighbourhood,

  3. $\pi $ is finite, and $\pi ^{-1}(\{ y\} ) = \{ x'\} $, and

  4. $\kappa (y)$ is a purely transcendental extension of $\kappa (s)$.

Moreover, if $f$ is locally of finite presentation then $\pi $ is of finite presentation.

Proof. The problem is local on $X$ and $S$, hence we may assume that $X$ and $S$ are affine. By Algebra, Lemma 10.125.3 after replacing $X$ by a standard open neighbourhood of $x$ in $X$ we may assume there is a factorization

\[ \xymatrix{ X \ar[r]^\pi & \mathbf{A}^ n_ S \ar[r] & S } \]

such that $\pi $ is quasi-finite and such that $\kappa (\pi (x))$ is purely transcendental over $\kappa (s)$. By Lemma 37.40.1 there exists an elementary étale neighbourhood

\[ (Y, y) \to (\mathbf{A}^ n_ S, \pi (x)) \]

and an open $X' \subset X \times _{\mathbf{A}^ n_ S} Y$ which contains a unique point $x'$ lying over $y$ such that $X' \to Y$ is finite. This proves (1) – (4) hold. For the final assertion, use Morphisms, Lemma 29.21.11. $\square$


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