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A dominant morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s0844801.png" /> between irreducible algebraic varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s0844802.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s0844803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s0844804.png" />, for which the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s0844805.png" /> is a [[Separable extension|separable extension]] of the subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s0844806.png" /> (isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s0844807.png" /> in view of the dominance). Non-separable mappings exist only when the characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s0844808.png" /> of the ground field is larger than 0. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s0844809.png" /> is a finite dominant morphism and its degree is not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448010.png" />, then it is separable. For a separable mapping there exists a non-empty open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448011.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448012.png" /> the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448014.png" /> surjectively maps the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448015.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448016.png" />, and conversely: If the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448018.png" /> are non-singular and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448019.png" /> is surjective, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448020.png" /> is a separable mapping.
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A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448021.png" /> of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448023.png" /> is called separated if the diagonal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448024.png" /> is closed. A composite of separated morphisms is separated; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448025.png" /> is separated if and only if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448026.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448027.png" /> such that the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084480/s08448028.png" /> is separated. A morphism of affine schemes is always separated. There are conditions for Noetherian schemes to be separated.
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A dominant morphism  $  f $
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between irreducible algebraic varieties  $  X $
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and  $  Y $,
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$  f:  X \rightarrow Y $,
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for which the field  $  K ( X) $
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is a [[Separable extension|separable extension]] of the subfield  $  f ^ { * } K ( Y) $(
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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 $
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the differential  $  ( df  ) _ {x} $
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of  $  f $
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surjectively maps the tangent space  $  T _ {X,x} $
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into  $  T _ {Y, f ( x) }  $,
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and conversely: If the points  $  x $
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and  $  f ( x) $
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are non-singular and  $  ( df  ) _ {x} $
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is surjective, then  $  f $
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is a separable mapping.
  
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A morphism  $  f:  X \rightarrow Y $
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of schemes  $  X $
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and  $  Y $
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is called separated if the diagonal in  $  X \times _ {Y} X $
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is closed. A composite of separated morphisms is separated;  $  f:  X \rightarrow Y $
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is separated if and only if for any point  $  y \in Y $
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there is a neighbourhood  $  V \ni y $
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such that the morphism  $  f:  f ^ { - 1 } ( V) \rightarrow V $
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is separated. A morphism of affine schemes is always separated. There are conditions for Noetherian schemes to be separated.
  
 
====Comments====
 
====Comments====
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" />.
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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 <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.
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Let $  A  ^ {1} $
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be the affine plane, and put $  U = A  ^ {1} \setminus  \{ ( 0, 0) \} $.  
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Let $  X $
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be obtained by glueing two copies of $  A  ^ {1} $
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along $  U $
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by the identity. Then $  X $
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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
How to Cite This Entry:
Separable mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_mapping&oldid=48669
This article was adapted from an original article by A.N. Rudakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article