The Stacks project

Lemma 10.43.5. Let $k$ be a field. Let $S$ be a geometrically reduced $k$-algebra. Let $R$ be any reduced $k$-algebra. Then $R \otimes _ k S$ is reduced.

Proof. By Lemma 10.43.4 we may assume that $R$ is of finite type over $k$. Then $R$, as a reduced Noetherian ring, embeds into a finite product of fields (see Lemmas 10.25.4, 10.31.6, and 10.25.1). Hence we may assume $R$ is a finite product of fields. In this case it follows from Definition 10.43.1 that $R \otimes _ k S$ is reduced. $\square$

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