Lemma 76.25.1 (082E)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.25: Normalization revisited
Lemma 76.25.1 (cite)

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Lemma 76.25.1. Let $S$ be a scheme. Let $f : Y \to X$ be a smooth morphism of algebraic spaces over $S$ . 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. By our construction of the integral closure, see Morphisms of Spaces, Definition 67.48.2, this reduces immediately to the case where $X$ and $Y$ are affine. In this case the result is Algebra, Lemma 10.147.4. $\square$

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