The Stacks project

Lemma 37.47.1. 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 37.41.1 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 052E. Beware of the difference between the letter 'O' and the digit '0'.