33.41 Normalization of one dimensional schemes
The normalization morphism of a Noetherian scheme of dimension $1$ has unexpectedly good properties by the Krull-Akizuki result.
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.
$\square$
Of course there is a variant of the following lemma in case $X$ is not reduced.
Lemma 33.41.2. Let $X$ be a reduced Nagata scheme of dimension $1$. Let $\nu : X^\nu \to X$ be the normalization. Let $x \in X$ denote a closed point. Then
$\nu : X^\nu \to X$ is finite, surjective, and birational,
$\mathcal{O}_ X \subset \nu _*\mathcal{O}_{X^\nu }$ and $\nu _*\mathcal{O}_{X^\nu }/\mathcal{O}_ X$ is a direct sum of skyscraper sheaves $\mathcal{Q}_ x$ in the singular points $x$ of $X$,
$A' = (\nu _*\mathcal{O}_{X^\nu })_ x$ is the integral closure of $A = \mathcal{O}_{X, x}$ in its total ring of fractions,
$\mathcal{Q}_ x = A'/A$ has finite length equal to the $\delta $-invariant of $X$ at $x$,
$A'$ is a semi-local ring which is a finite product of Dedekind domains,
$A^\wedge $ is a reduced Noetherian complete local ring of dimension $1$,
$(A')^\wedge $ is the integral closure of $A^\wedge $ in its total ring of fractions,
$(A')^\wedge $ is a finite product of complete discrete valuation rings, and
$A'/A \cong (A')^\wedge /A^\wedge $.
Proof.
We may and will use all the results of Lemma 33.41.1. Finiteness of $\nu $ follows from Morphisms, Lemma 29.54.11. Since $X$ is reduced, Nagata, of dimension $1$, we see that the regular locus is a dense open $U \subset X$ by More on Algebra, Proposition 15.48.7. Since a regular scheme is normal, this shows that $\nu $ is an isomorphism over $U$. Since $\dim (X) \leq 1$ this implies that $\nu $ is not an isomorphism over a discrete set of closed points $x \in X$. In particular we see that we have a short exact sequence
\[ 0 \to \mathcal{O}_ X \to \nu _*\mathcal{O}_{X^\nu } \to \bigoplus \nolimits _{x \in X \setminus U} \mathcal{Q}_ x \to 0 \]
As we have the description of the stalks of $\nu _*\mathcal{O}_{X^\nu }$ by Lemma 33.41.1, we conclude that $Q_ x = A'/A$ indeed has length equal to the $\delta $-invariant of $X$ at $x$. Note that $Q_ x \not= 0$ exactly when $x$ is a singular point for example by Lemma 33.39.4. The description of $A'$ as a product of semi-local Dedekind domains follows from Lemma 33.41.1 as well. The relationship between $A$, $A'$, and $(A')^\wedge $ we have see in Lemma 33.39.5 (and its proof).
$\square$
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