The Stacks project

Lemma 37.41.4. 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 37.41.2. Since $X_ U \to U$ is separated (Schemes, Lemma 26.21.12) and $V_ i \to U$ is finite hence proper (Morphisms, Lemma 29.44.11) we see that $V_ i \subset X_ U$ is closed by Morphisms, Lemma 29.41.7. 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. $\square$

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Your email address will not be published.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