Lemma 10.45.6 (00I4)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.45: Perfect fields
Lemma 10.45.6 (cite)

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Lemma 10.45.6. Let $k$ be a perfect field. Any reduced $k$ algebra is geometrically reduced over $k$ . Let $R$ , $S$ be $k$ -algebras. Assume both $R$ and $S$ are reduced. Then the $k$ -algebra $R \otimes _ k S$ is reduced.

Proof. The first statement follows from Lemma 10.44.4. For the second statement use the first statement and Lemma 10.43.5. $\square$

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