Lemma 10.36.13 (02JK)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.36: Finite and integral ring extensions
Lemma 10.36.13 (cite)

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Integrality and finiteness are preserved under base change.

Lemma 10.36.13. Let $R \to S$ and $R \to R'$ be ring maps. Set $S' = R' \otimes _ R S$.

If $R \to S$ is integral so is $R' \to S'$.
If $R \to S$ is finite so is $R' \to S'$.

Proof. We prove (1). Let $s_ i \in S$ be generators for $S$ over $R$. Each of these satisfies a monic polynomial equation $P_ i$ over $R$. Hence the elements $1 \otimes s_ i \in S'$ generate $S'$ over $R'$ and satisfy the corresponding polynomial $P_ i'$ over $R'$. Since these elements generate $S'$ over $R'$ we see that $S'$ is integral over $R'$. Proof of (2) omitted. $\square$

Comments (1)

Comment #1092 by Alex Youcis on October 17, 2014 at 07:08

Suggested slogan: Integrality and finitneness are preserved under base change.

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