Lemma 37.44.3 (0AH8)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.44: Applications of Zariski's Main Theorem, I
Lemma 37.44.3 (cite)

previous lemma

numberstags

Lemma 37.44.3. Consider a commutative diagram of schemes

$\xymatrix{ X \ar[rr]_ h \ar[rd]_ f & & Y \ar[ld]^ g \\ & S }$

Let $s \in S$ . Assume

$X \to S$ is a proper morphism,
$Y \to S$ is separated and locally of finite type, and
the image of $X_ s \to Y_ s$ is finite.

Then there is an open subspace $U \subset S$ containing $s$ such that $X_ U \to Y_ U$ factors through a closed subscheme $Z \subset Y_ U$ finite over $U$ .

Proof. Let $Z \subset Y$ be the scheme theoretic image of $h$ , see Morphisms, Section 29.6. By Morphisms, Lemma 29.41.10 the morphism $X \to Z$ is surjective and $Z \to S$ is proper. Thus $X_ s \to Z_ s$ is surjective. We see that either (3) implies $Z_ s$ is finite. Hence $Z \to S$ is finite in an open neighbourhood of $s$ by Lemma 37.44.2. $\square$

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