Lemma 31.25.2 (035T)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.25: Meromorphic functions and sections; reduced case
Lemma 31.25.2 (cite)

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Lemma 31.25.2. Let $X$ be a scheme. Assume $X$ is reduced and any quasi-compact open $U \subset X$ has a finite number of irreducible components. Then the normalization morphism $\nu : X^\nu \to X$ is the morphism

\[ \underline{\mathop{\mathrm{Spec}}}_ X(\mathcal{O}') \longrightarrow X \]

where $\mathcal{O}' \subset \mathcal{K}_ X$ is the integral closure of $\mathcal{O}_ X$ in the sheaf of meromorphic functions.

Proof. Compare the definition of the normalization morphism $\nu : X^\nu \to X$ (see Morphisms, Definition 29.54.1) with the description of $\mathcal{K}_ X$ in Lemma 31.25.1 above. $\square$

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