The normalization of an irreducible affine variety

Assume the Axiom of Choice. Let $k$ be an algebraically closed field and let $X$ be an irreducible affine variety over $k$ with coordinate ring $A=k[X]$ and function field $k(X)=\mathrm{Frac}(A)$ (A classical affine variety, The function field of an irreducible classical affine variety).

Let $B\subseteq k(X)$ be the integral closure of $A$ in $k(X)$ (Integral closure in an extension ring and integrally closed domains). Then $B$ is a finite $A$-module by A finite-type domain over a field has finite normalization and is an integrally closed domain by The integral closure of a domain in a field extension is integrally closed.

Concretely, $B$ is a reduced affine $k$-algebra: it is a domain, it is finitely generated as a $k$-algebra because it is a finite module over the finitely generated $k$-algebra $A$, and it is reduced because it is a domain. By the object-level duality of Affine algebraic sets and reduced affine k-algebras at the object level there is an affine algebraic set $X^{\nu }\subseteq {\mathbf{A}}_k^m$ with $k[X^{\nu }]\cong B$; since $B$ is a domain, $X^{\nu }$ is irreducible, hence a classical affine variety (A classical affine variety).

The normalization of $X$ is the pair $(X^{\nu },\nu )$, where $\nu \text{\hspace{0.17em}⁣}:X^{\nu }\to X$ is the morphism corresponding under the coordinate-ring anti-equivalence (Affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms, Morphisms of classical affine varieties) to the inclusion of $k$-algebras $A\hookrightarrow B$; the inclusion presents $B$ as a finite $A$-module, so $\nu$ is a finite morphism (Finite morphisms of classical varieties). All points of $X^{\nu }$ are normal: for $y\in X^{\nu }$ the local ring ${\mathcal{O}}_{X^{\nu },y}$ is a localisation of $B$ at a maximal ideal, and $B$ is an integrally closed domain, so every such localisation is an integrally closed domain as well; hence $X^{\nu }$ is a normal variety (Normal points and normal varieties).

Recorded property. The universal property of the normalization, and with it the fact that the pair $(X^{\nu },\nu )$ is unique up to a unique isomorphism over $X$, is proved later on this page. The construction above is the affine case of the normalization of a variety in a finite extension of its function field; finiteness and birationality of $\nu$ are properties of this construction, not additional data.