Lemma 10.123.10 (00Q2)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.123: Zariski's Main Theorem
Lemma 10.123.10 (cite)

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Lemma 10.123.10. Suppose $R \subset S$ is an inclusion of reduced rings. Assume $x \in S$ be strongly transcendental over $R$, and $S$ finite over $R[x]$. Then $R \to S$ is not quasi-finite at any prime of $S$.

Proof. Let $\mathfrak q \subset S$ be any prime. Choose a minimal prime $\mathfrak q' \subset \mathfrak q$. According to Lemmas 10.123.8 and 10.123.9 the extension $R/(R \cap \mathfrak q') \subset S/\mathfrak q'$ is not quasi-finite at the prime corresponding to $\mathfrak q$. By Lemma 10.122.6 the extension $R \to S$ is not quasi-finite at $\mathfrak q$. $\square$

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