Namespaces
Variants
Actions

Constructible subset

From Encyclopedia of Mathematics
Revision as of 17:11, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

of an algebraic variety

A finite union of locally closed (in the Zariski topology) subsets. A locally closed subset is, by definition, an intersection of an open and a closed subset. The constructible subsets form a Boolean algebra and can be defined as elements of the Boolean algebra generated by the algebraic subvarieties. The role of constructible subsets in algebraic geometry is revealed by Chevalley's theorem: If is a morphism of algebraic varieties, then (and, moreover, the image of any constructible subset in ) is a constructible subset in . This is related to the fact that "algebraic" conditions determine the constructible subsets of an algebraic variety.

A mapping is called constructible if is finite and if for any point the pre-image is a constructible subset in .

References

[1] A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" , I. Le langage des schémes , Springer (1971)
[2] A. Borel, "Linear algebraic groups" , Benjamin (1969)
How to Cite This Entry:
Constructible subset. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Constructible_subset&oldid=15045
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article