Lemma 15.104.2. Let $A \to B$ be a ring map such that $B \otimes _ A B \to B$ is flat. Let $N$ be a $B$-module. If $N$ is flat as an $A$-module, then $N$ is flat as a $B$-module.
Proof. Assume $N$ is a flat as an $A$-module. Then the functor
is exact. As $B \otimes _ A B \to B$ is flat we conclude that the functor
is exact, hence $N$ is flat over $B$. $\square$
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