The Stacks project

Lemma 10.147.5. Let $R \to S$ and $R \to B$ be ring maps. Let $A \subset B$ be the integral closure of $R$ in $B$. Let $A' \subset S \otimes _ R B$ be the integral closure of $S$ in $S \otimes _ R B$. If $S$ is a filtered colimit of smooth $R$-algebras, then the canonical map $S \otimes _ R A \to A'$ is an isomorphism.

Proof. This follows from the straightforward fact that taking tensor products and taking integral closures commutes with filtered colimits and Lemma 10.147.4. $\square$

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