Separable mapping

A dominant morphism between irreducible algebraic varieties and , , for which the field is a separable extension of the subfield (isomorphic to in view of the dominance). Non-separable mappings exist only when the characteristic of the ground field is larger than 0. If is a finite dominant morphism and its degree is not divisible by , then it is separable. For a separable mapping there exists a non-empty open set such that for all the differential of surjectively maps the tangent space into , and conversely: If the points and are non-singular and is surjective, then is a separable mapping.

A morphism of schemes and is called separated if the diagonal in is closed. A composite of separated morphisms is separated; is separated if and only if for any point there is a neighbourhood such that the morphism is separated. A morphism of affine schemes is always separated. There are conditions for Noetherian schemes to be separated.

Comments

A morphism of algebraic varieties or schemes is called dominant if is dense in .

In the Russian literature the phrase "separabel'noe otobrazhenie" which literally translates as "separable mapping" is sometimes encountered in the meaning "separated mapping" .

Let be the affine plane, and put . Let be obtained by glueing two copies of along by the identity. Then is a non-separated scheme.

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2