The Stacks project

Lemma 37.41.1. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Set $s = f(x)$. Assume that

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

  2. $x \in X_ s$ is isolated1.

Then there exist

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

  2. an open subscheme $V \subset X_ U$ (see

such that

  1. $V \to U$ is a finite morphism,

  2. there is a unique point $v$ of $V$ mapping to $u$ in $U$, and

  3. the point $v$ maps to $x$ under the morphism $X_ U \to X$, inducing $\kappa (x) = \kappa (v)$.

Moreover, for any elementary étale neighbourhood $(U', u') \to (U, u)$ setting $V' = U' \times _ U V \subset X_{U'}$ the triple $(U', u', V')$ satisfies the properties (i), (ii), and (iii) as well.

Proof. Let $Y \subset X$, $W \subset S$ be affine opens such that $f(Y) \subset W$ and such that $x \in Y$. Note that $x$ is also an isolated point of the fibre of the morphism $f|_ Y : Y \to W$. If we can prove the theorem for $f|_ Y : Y \to W$, then the theorem follows for $f$. Hence we reduce to the case where $f$ is a morphism of affine schemes. This case is Algebra, Lemma 10.145.2. $\square$

[1] In the presence of (1) this means that $f$ is quasi-finite at $x$, see Morphisms, Lemma 29.20.6.

Comments (0)

There are also:

  • 2 comment(s) on Section 37.41: Étale localization of quasi-finite morphisms

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Your email address will not be published.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 02LK. Beware of the difference between the letter 'O' and the digit '0'.