The Stacks project

Lemma 41.17.1. 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 $f$ is unramified at each $x_ i$. Then there exists an étale neighbourhood $(U, u) \to (S, s)$ and opens $V_{i, j} \subset X_ U$, $i = 1, \ldots , n$, $j = 1, \ldots , m_ i$ such that

  1. $V_{i, j} \to U$ is a closed immersion passing through $u$,

  2. $u$ is not in the image of $V_{i, j} \cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and

  3. any point of $(X_ U)_ u$ mapping to $x_ i$ is in some $V_{i, j}$.

Proof. By Morphisms, Definition 29.35.1 there exists an open neighbourhood of each $x_ i$ which is locally of finite type over $S$. Replacing $X$ by an open neighbourhood of $\{ x_1, \ldots , x_ n\} $ we may assume $f$ is locally of finite type. Apply More on Morphisms, Lemma 37.41.3 to get the étale neighbourhood $(U, u)$ and the opens $V_{i, j}$ finite over $U$. By Lemma 41.7.3 after possibly shrinking $U$ we get that $V_{i, j} \to U$ is a closed immersion. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 41.17: Étale local structure of unramified 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Your email address will not be published.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 04HH. Beware of the difference between the letter 'O' and the digit '0'.