View source for Complete algebraic variety

A generalization of the concept of a compact complex algebraic variety. A separated variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237101.png" /> is called complete if for any variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237102.png" /> the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237103.png" /> is a closed morphism, i.e. it maps closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237104.png" /> (in the Zariski topology) into closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237105.png" />. There is what is called the valuative completeness criterion: For any discrete valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237106.png" /> with field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237107.png" /> and any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237108.png" /> there should be a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237109.png" /> that extends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371010.png" />. This condition is an analogue of the requirement that any sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371011.png" /> has a limit point.

Any projective variety is complete, but not vice versa. For any complete algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371012.png" /> there exists a projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371013.png" /> and a projective birational morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371014.png" /> (Chow's lemma). For any algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371015.png" /> there exists an open imbedding into a complete variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371016.png" /> (Nagata's theorem). A generalization of the concept of a complete algebraic variety to the relative case is that of a proper morphism of schemes.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.R. Shafarevich,   "Basic algebraic geometry" , Springer  (1977)  (Translated from Russian)</TD></TR></table>