Lemma 37.43.2 (02LR)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.43: Zariski's Main Theorem
Lemma 37.43.2 (cite)

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Quasi-finite, separated morphisms are quasi-affine

Lemma 37.43.2. Let $f : X \to S$ be a morphism of schemes. Assume $f$ is quasi-finite and separated. Let $S'$ be the normalization of $S$ in $X$ , see Morphisms, Definition 29.53.3. Picture:

$\xymatrix{ X \ar[rd]_ f \ar[rr]_{f'} & & S' \ar[ld]^\nu \\ & S & }$

Then $f'$ is a quasi-compact open immersion and $\nu$ is integral. In particular $f$ is quasi-affine.

Proof. This follows from Lemma 37.43.1. Namely, by that lemma there exists an open subscheme $U' \subset S'$ such that $(f')^{-1}(U') = X$ and $X \to U'$ is an isomorphism. In other words, $f'$ is an open immersion. Note that $f'$ is quasi-compact as $f$ is quasi-compact and $\nu : S' \to S$ is separated (Schemes, Lemma 26.21.14). It follows that $f$ is quasi-affine by Morphisms, Lemma 29.13.3. $\square$

Comments (2)

Comment #5119 by slogan_bot on May 28, 2020 at 02:36

Suggested slogan: Quasi-finite, separated morphisms are quasi-affine

Also, why is there an "(!)" in the proof?

Comment #5326 by Johan on June 19, 2020 at 11:37

Thanks and fixed here.

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