The Stacks project

Lemma 15.104.8. Let $A \to B$ be a ring map such that $B \otimes _ A B \to B$ is flat.

  1. If $A$ is an absolutely flat ring, then so is $B$.

  2. If $A$ is reduced and $A \to B$ is weakly étale, then $B$ is reduced.

Proof. Part (1) follows immediately from Lemma 15.104.2 and the definitions. If $A$ is reduced, then there exists an injection $A \to A' = \prod _{\mathfrak p \subset A\text{ minimal}} A_\mathfrak p$ of $A$ into an absolutely flat ring (Algebra, Lemma 10.25.2 and Lemma 15.104.6). If $A \to B$ is flat, then the induced map $B \to B' = B \otimes _ A A'$ is injective too. By Lemma 15.104.7 the ring map $A' \to B'$ is weakly étale. By part (1) we see that $B'$ is absolutely flat. By Lemma 15.104.5 the ring $B'$ is reduced. Hence $B$ is reduced. $\square$

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