Lemma 33.41.1 (0C45)—The Stacks project

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Part 2: Schemes
Chapter 33: Varieties
Section 33.41: Normalization of one dimensional schemes
Lemma 33.41.1 (cite)

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Lemma 33.41.1. Let $X$ be a locally Noetherian scheme of dimension $1$. Let $\nu : X^\nu \to X$ be the normalization. Then

$\nu $ is integral, surjective, and induces a bijection on irreducible components,
there is a factorization $X^\nu \to X_{red} \to X$ and the morphism $X^\nu \to X_{red}$ is the normalization of $X_{red}$,
$X^\nu \to X_{red}$ is birational,
for every closed point $x \in X$ the stalk $(\nu _*\mathcal{O}_{X^\nu })_ x$ is the integral closure of $\mathcal{O}_{X, x}$ in the total ring of fractions of $(\mathcal{O}_{X, x})_{red} = \mathcal{O}_{X_{red}, x}$,
the fibres of $\nu $ are finite and the residue field extensions are finite,
$X^\nu $ is a disjoint union of integral normal Noetherian schemes and each affine open is the spectrum of a finite product of Dedekind domains.

Proof. Many of the results are in fact general properties of the normalization morphism, see Morphisms, Lemmas 29.54.2, 29.54.4, 29.54.5, and 29.54.7. What is not clear is that the fibres are finite, that the induced residue field extensions are finite, and that $X^\nu $ locally looks like the spectrum of a Dedekind domain (and hence is Noetherian). To see this we may assume that $X = \mathop{\mathrm{Spec}}(A)$ is affine, Noetherian, dimension $1$, and that $A$ is reduced. Then we may use the description in Morphisms, Lemma 29.54.3 to reduce to the case where $A$ is a Noetherian domain of dimension $1$. In this case the desired properties follow from Krull-Akizuki in the form stated in Algebra, Lemma 10.120.18. $\square$

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