(Normal)
An integral domain is normal if it is integrally closed in its fraction field. A scheme \(X\) is normal if every local ring \(\OO_{X,p}\), for \(p \in X\), is a normal domain.
A regular local ring is normal, and the converse fails in dimension \(\geq 2\). In dimension \(1\) the two agree: a Noetherian local domain of dimension \(1\) is normal exactly when it is a discrete valuation ring, that is, regular.
(Normalization)
Let \(X\) be an integral scheme. A normalization of \(X\) is a normal integral scheme \(\widetilde{X}\) with a dominant morphism \(\nu \colon \widetilde{X} \to X\) such that every dominant morphism \(f \colon Y \to X\) from a normal integral scheme \(Y\) factors uniquely as \(f = \nu \circ \tilde{f}\) for a morphism \(\tilde{f} \colon Y \to \widetilde{X}\).
(Existence of the normalization)
Every integral scheme \(X\) has a normalization, unique up to unique isomorphism. If \(X = \Spec A\) is affine, \(\widetilde{X} = \Spec \widetilde{A}\) for the integral closure \(\widetilde{A}\) of \(A\) in its fraction field, and in general \(\widetilde{X}\) is obtained by gluing these over an affine cover. If \(X\) is a variety over a field \(k\), then \(\nu\) is finite and birational, and \(\widetilde{X}\) is projective when \(X\) is projective [@Har10a, Exercise II.3.8].
The normalization of an integral curve is regular, hence smooth over a perfect field. Thus normalization resolves singularities of curves over perfect fields.
The quadric cone \(V(xy - z^2) \subseteq \AA^3\) is normal and singular at the origin: it is regular in codimension one and Cohen–Macaulay, so it is normal by Serre’s criterion \(R_1 + S_2\). The cuspidal cubic, with coordinate ring \(k[t^2,t^3] \subseteq k[t]\), is not normal, and its normalization is \(\Spec k[t]\).
Dependencies
D-0SYCYSmooth and singular points; Jacobian and regularity criteriauses
Backlinks
D-5PQ5WWeil and Cartier divisors, and when they agreeusesPR-DIVZEROPOLEZeros and poles of a rational functionuses