Difference between revisions of "Separable mapping"
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+ | 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|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==== | ====Comments==== | ||
− | A morphism | + | 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" . | In the Russian literature the phrase "separabel'noe otobrazhenie" which literally translates as "separable mapping" is sometimes encountered in the meaning "separated mapping" . | ||
− | Let | + | 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==== | ====References==== | ||
<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> | <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> |
Latest revision as of 08:13, 6 June 2020
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 |
Separable mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_mapping&oldid=23975