Lemma 37.40.6. Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that

1. $f$ is locally of finite type,

2. $f$ is separated, and

3. $X_ s$ has at most finitely many isolated points.

Then there exists an elementary étale neighbourhood $(U, u) \to (S, s)$ and a decomposition

$U \times _ S X = W \amalg V$

into open and closed subschemes such that the morphism $V \to U$ is finite, and the fibre $W_ u$ of the morphism $W \to U$ contains no isolated points. In particular, if $f^{-1}(s)$ is a finite set, then $W_ u = \emptyset$.

Proof. This is clear from Lemma 37.40.4 by choosing $x_1, \ldots , x_ n$ the complete set of isolated points of $X_ s$ and setting $V = \bigcup V_ i$. $\square$

There are also:

• 2 comment(s) on Section 37.40: Étale localization of quasi-finite morphisms

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).