Lemma 41.14.2 (04DH)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 41: Étale Morphisms of Schemes
Section 41.14: Topological properties of étale morphisms
Lemma 41.14.2 (cite)

numberstags

Lemma 41.14.2. Let $\pi : X \to S$ be a morphism of schemes. Let $s \in S$ . Assume that

$\pi$ is finite,
$\pi$ is étale,
$\pi ^{-1}(\{ s\} ) = \{ x\}$ , and
$\kappa (s) \subset \kappa (x)$ is purely inseparable ¹.

Then there exists an open neighbourhood $U$ of $s$ such that $\pi |_{\pi ^{-1}(U)} : \pi ^{-1}(U) \to U$ is an isomorphism.

Proof. By Lemma 41.7.3 there exists an open neighbourhood $U$ of $s$ such that $\pi |_{\pi ^{-1}(U)} : \pi ^{-1}(U) \to U$ is a closed immersion. But a morphism which is étale and a closed immersion is an open immersion (for example by Theorem 41.14.1). Hence after shrinking $U$ we obtain an isomorphism. $\square$

[1] In view of condition (2) this is equivalent to

$\kappa (s) = \kappa (x)$ .

Comments (0)

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).

Comments (0)

Post a comment