Lemma 10.35.13. 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 $R' \to S'$.

** Integrality and finiteness are preserved under base change. **

Lemma 10.35.13. 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.
$\square$

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## Comments (1)

Comment #1092 by Alex Youcis on