Lemma 37.18.1. Let $f : Y \to X$ be a smooth morphism of schemes. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras. The integral closure of $\mathcal{O}_ Y$ in $f^*\mathcal{A}$ is equal to $f^*\mathcal{A}'$ where $\mathcal{A}' \subset \mathcal{A}$ is the integral closure of $\mathcal{O}_ X$ in $\mathcal{A}$.

Proof. This is a translation of Algebra, Lemma 10.147.4 into the language of schemes. Details omitted. $\square$

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