Difference between revisions of "Separable mapping"
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A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448029.png" /> of algebraic varieties or schemes is called dominant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448030.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448031.png" />. | A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448029.png" /> of algebraic varieties or schemes is called dominant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448030.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448031.png" />. | ||
− | In the Russian literature the phrase | + | In the Russian literature the phrase "separabel'noe otobrazhenie" which literally translates as "separable mapping" is sometimes encountered in the meaning "separated mapping" . |
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448032.png" /> be the affine plane, and put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448033.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448034.png" /> be obtained by glueing two copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448035.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448036.png" /> by the identity. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448037.png" /> is a non-separated scheme. | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448032.png" /> be the affine plane, and put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448033.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448034.png" /> be obtained by glueing two copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448035.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448036.png" /> by the identity. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448037.png" /> is a non-separated scheme. | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table> |
Revision as of 21:56, 30 March 2012
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 MR0463157 Zbl 0367.14001 |
Separable mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_mapping&oldid=23975