The Stacks project

Lemma 15.104.22. Let $A$ be a normal domain with fraction field $K$. Let $A \to B$ be weakly étale. Then $B$ is integrally closed in $B \otimes _ A K$.

Proof. Choose a diagram as in Lemma 15.104.20. As $A \to B$ is flat, the base change gives a cartesian diagram

\[ \xymatrix{ B \ar[d] \ar[r] & B \otimes _ A K \ar[d] \\ B \otimes _ A V \ar[r] & B \otimes _ A L } \]

of rings. Note that $V \to B \otimes _ A V$ is weakly étale (Lemma 15.104.7), hence $B \otimes _ A V$ has weak dimension at most $1$ by Lemma 15.104.4. Note that $B \otimes _ A V \to B \otimes _ A L$ is a flat, injective, epimorphism of rings as a flat base change of such (Algebra, Lemmas 10.39.7 and 10.107.3). By Lemma 15.104.21 we see that $B \otimes _ A V$ is integrally closed in $B \otimes _ A L$. It follows from the cartesian property of the diagram that $B$ is integrally closed in $B \otimes _ A K$. $\square$

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