View source for Separable mapping

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.

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.



====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" />.

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.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)  pp. Sect. IV.2</TD></TR></table>