Lemma 29.54.3 (035P)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.54: Normalization
Lemma 29.54.3 (cite)

Lemma 29.54.3. Let $X$ be a reduced scheme such that every quasi-compact open has finitely many irreducible components. Let $\mathop{\mathrm{Spec}}(A) = U \subset X$ be an affine open. Then

$A$ has finitely many minimal primes $\mathfrak q_1, \ldots , \mathfrak q_ t$,
the total ring of fractions $Q(A)$ of $A$ is $Q(A/\mathfrak q_1) \times \ldots \times Q(A/\mathfrak q_ t)$,
the integral closure $A'$ of $A$ in $Q(A)$ is the product of the integral closures of the domains $A/\mathfrak q_ i$ in the fields $Q(A/\mathfrak q_ i)$, and
$\nu ^{-1}(U)$ is identified with the spectrum of $A'$ where $\nu : X^\nu \to X$ is the normalization morphism.

Proof. Minimal primes correspond to irreducible components (Algebra, Lemma 10.26.1), hence we have (1) by assumption. Then $(0) = \mathfrak q_1 \cap \ldots \cap \mathfrak q_ t$ because $A$ is reduced (Algebra, Lemma 10.17.2). Then we have $Q(A) = \prod A_{\mathfrak q_ i} = \prod \kappa (\mathfrak q_ i)$ by Algebra, Lemmas 10.25.4 and 10.25.1. This proves (2). Part (3) follows from Algebra, Lemma 10.37.16, or Lemma 29.53.10. Part (4) holds because it is clear that $f^{-1}(U) \to U$ is the morphism

\[ \mathop{\mathrm{Spec}}\left(\prod \kappa (\mathfrak q_ i)\right) \longrightarrow \mathop{\mathrm{Spec}}(A) \]

where $f : Y \to X$ is the morphism (29.54.0.1). $\square$

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