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 |
Separable mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_mapping&oldid=17689