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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