An integral locally Noetherian scheme is normal when every local ring is an integrally closed domain.
More generally a scheme is normal when every local ring is an integrally closed domain; the scheme need not be globally integral, for example a disjoint union of normal integral components.
For an integral scheme, take on each affine chart the integral closure of its coordinate ring in its function field. Integral closure commutes with localization, so these rings glue to a normal scheme with an integral birational morphism to the original scheme. This is the normalization. It need not be finite without additional hypotheses; for an algebraic curve of finite type over a field it is finite. It is an isomorphism exactly when the original scheme is normal.
An algebraic variety whose local rings are integrally closed domains. In particular an irreducible variety that is a normal variety has normal affine coordinate rings, enabling codimension-two extension of regular functions on a normal variety.
A singular point on a normal two-dimensional algebraic variety has an integrally closed local domain which is not a regular local ring. A rational double point is an important isolated example with a crepant minimal resolution.
A rational double point is a normal surface singularity whose minimal resolution has a configuration of smooth rational -curves of type , or . Birational projective models of K3 surfaces arising from nef big systems contract orthogonal -curves to these singularities.
Over an algebraically closed field of characteristic different from two, a nondegenerate quadratic form of rank at least three is irreducible, its cone has only the vertex singular, and its hypersurface ring is Cohen-Macaulay. Thus and in the Serre criterion for normality hold. In characteristic two, the reduced zero variety of is instead the hyperplane , which is normal. The unreduced double-hyperplane scheme is not normal.
For an integral normal variety that is Noetherian, removing a closed subset of codimension at least two does not create new regular functions. Affine-locally, a normal Noetherian domain is the intersection of its localizations at height-one primes inside its fraction field. The complement still contains every height-one point, so a regular function on it lies in each such localization and therefore in the original ring. On projective space this also follows directly by excluding irreducible polynomial factor of the denominators on each polynomial affine chart.
A Noetherian ring is normal exactly when it satisfies , regularity in codimension one, and , depth at least at every localization.
