Normal scheme

In mathematics, in the field of algebraic geometry, a normal scheme is a scheme X for which every stalk (local ring)

O_X,x

of its structure sheaf O_X is an integrally closed local ring; that is, each stalk is an integral domain such that its integral closure in its field of fractions is equal to itself. (An older and unrelated meaning of normal is that a normal variety is a subvariety of projective space such that the linear system giving the embedding is complete: see rational normal surface and rational normal curve for examples.)

An example of a normal scheme is a regular scheme.

Any reduced scheme has a normalization, whose construction we first give for irreducible reduced schemes.

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 the initial scheme is not irreducible, one can define the normalization as the disjoint union of the normalizations of the irreducible components. An alternate, equivalent, definition uses integral closures in rings of fractions where any nonzero divisor is allowed in the denominator.

See also

• Integral closure
• Semi-normal scheme

References

• Template:Hartshorne AG, p. 91