# 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 |

**How to Cite This Entry:**

Separable mapping.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Separable_mapping&oldid=48669