Separable mapping

A dominant morphism $ f $ between irreducible algebraic varieties $ X $ and $ Y $, $ f: X \rightarrow Y $, for which the field $ K ( X) $ is a separable extension of the subfield $ f ^ { * } K ( Y) $( isomorphic to $ K ( Y) $ in view of the dominance). Non-separable mappings exist only when the characteristic $ p $ of the ground field is larger than 0. If $ f $ is a finite dominant morphism and its degree is not divisible by $ p $, then it is separable. For a separable mapping there exists a non-empty open set $ U \subset X $ such that for all $ x \in U $ the differential $ ( df ) _ {x} $ of $ f $ surjectively maps the tangent space $ T _ {X,x} $ into $ T _ {Y, f ( x) } $, and conversely: If the points $ x $ and $ f ( x) $ are non-singular and $ ( df ) _ {x} $ is surjective, then $ f $ is a separable mapping.

A morphism $ f: X \rightarrow Y $ of schemes $ X $ and $ Y $ is called separated if the diagonal in $ X \times _ {Y} X $ is closed. A composite of separated morphisms is separated; $ f: X \rightarrow Y $ is separated if and only if for any point $ y \in Y $ there is a neighbourhood $ V \ni y $ such that the morphism $ f: f ^ { - 1 } ( V) \rightarrow V $ is separated. A morphism of affine schemes is always separated. There are conditions for Noetherian schemes to be separated.

Comments

A morphism $ f: X \rightarrow Y $ of algebraic varieties or schemes is called dominant if $ f( X) $ is dense in $ Y $.

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

Let $ A ^ {1} $ be the affine plane, and put $ U = A ^ {1} \setminus \{ ( 0, 0) \} $. Let $ X $ be obtained by glueing two copies of $ A ^ {1} $ along $ U $ by the identity. Then $ X $ is a non-separated scheme.

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001