The Stacks project

Lemma 10.43.4. Let $k$ be a field. Let $R$, $S$ be $k$-algebras.

  1. If $R \otimes _ k S$ is nonreduced, then there exist finitely generated subalgebras $R' \subset R$, $S' \subset S$ such that $R' \otimes _ k S'$ is not reduced.

  2. If $R \otimes _ k S$ contains a nonzero zerodivisor, then there exist finitely generated subalgebras $R' \subset R$, $S' \subset S$ such that $R' \otimes _ k S'$ contains a nonzero zerodivisor.

  3. If $R \otimes _ k S$ contains a nontrivial idempotent, then there exist finitely generated subalgebras $R' \subset R$, $S' \subset S$ such that $R' \otimes _ k S'$ contains a nontrivial idempotent.

Proof. Suppose $z \in R \otimes _ k S$ is nilpotent. We may write $z = \sum _{i = 1, \ldots , n} x_ i \otimes y_ i$. Thus we may take $R'$ the $k$-subalgebra generated by the $x_ i$ and $S'$ the $k$-subalgebra generated by the $y_ i$. The second and third statements are proved in the same way. $\square$

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