# Normal scheme

A scheme all local rings (cf. Local ring) of which are normal (that is, reduced and integrally closed in their ring of fractions). A normal scheme is locally irreducible; for such a scheme the concepts of a connected component and an irreducible component are the same. The set of singular points of a Noetherian normal scheme has codimension greater than 1. The following normality criterion holds : A Noetherian scheme $X$ is normal if and only if two conditions are satisfied: 1) for any point $x \in X$ of codimension $\leq 1$ the local ring ${\mathcal O} _ {X,x}$ is regular (cf. Regular ring (in commutative algebra)); and 2) for any point $x \in X$ of codimension $> 1$ the depth of the ring (cf. Depth of a module) ${\mathcal O} _ {X,x}$ is greater than 1. Every reduced scheme $X$ has a normal scheme $X ^ \nu$ canonically connected with it (normalization). The $X$- scheme $X ^ \nu$ is integral, but not always finite over $X$. However, if $X$ is excellent (see Excellent ring), for example, if $X$ is a scheme of finite type over a field, then $X ^ \nu$ is finite over $X$.