Loading [MathJax]/extensions/tex2jax.js

The Stacks project

Lemma 41.18.2. 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 separated and $f$ is étale at each $x_ i$. Then there exists an étale neighbourhood $(U, u) \to (S, s)$ and a finite disjoint union decomposition

\[ X_ U = W \amalg \coprod \nolimits _{i, j} V_{i, j} \]

of schemes such that

  1. $V_{i, j} \to U$ is an isomorphism,

  2. the fibre $W_ u$ contains no point mapping to any $x_ i$.

In particular, if $f^{-1}(\{ s\} ) = \{ x_1, \ldots , x_ n\} $, then the fibre $W_ u$ is empty.

Proof. An étale morphism is unramified, hence we may apply Lemma 41.17.2. As in the proof of Lemma 41.18.1 the morphisms $V_{i, j} \to U$ are open immersions and we win after replacing $U$ by the intersection of their images. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 41.18: Étale local structure of étale morphisms

Your email address will not be published.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.