Normal scheme

A normal scheme is a scheme where every stalk (local ring) is an integrally closed local ring, that is each stalk is a domain such that it's integral closure in it's fraction field is equal to itself.

Any reduced scheme can be normalized. This is done by first separating the scheme into it's irreducible components and normalizing each of them individually.

An irreducible and reduced scheme $X$ has the property that every affine chart is a domain. Choose an affine cover corresponding to rings $A_{i}$. Compute the integral closure of each of these in its fraction field, denote them by ${\overline {A_{i}}}$. It is not hard to see that one can construct a new scheme ${\overline {X}}$ by gluing together the affine schemes Spec${\overline {A_{i}}}$.

If your initial scheme was not irreducible, it's normalization is just equal to the disjoint union of the normalizations of the irreducible components.

See also

• Integral Closure
• Semi-normal scheme