Difference between revisions of "Constructible subset"
From Encyclopedia of Mathematics
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''of an algebraic variety'' | ''of an algebraic variety'' | ||
| − | A finite union of locally closed (in the [[ | + | 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 $f:X\rightarrow Y$ is a morphism of algebraic varieties, then $f(X)$ (and, moreover, the image under $f$ of any constructible subset in $X$) is a constructible subset in $Y$. This is related to the fact that "algebraic" conditions determine the constructible subsets of an algebraic variety. |
| − | A mapping | + | A mapping $h:X\rightarrow T$ is called constructible if $h(X)$ is finite and if for any point $t\in T$ the pre-image $h^{-1}(t)$ is a constructible subset in $X$. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique" , '''I. Le langage des schémas''' , Springer (1971) {{MR|0217085}} {{ZBL|0203.23301}} </TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Latest revision as of 06:07, 16 July 2024
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 $f:X\rightarrow Y$ is a morphism of algebraic varieties, then $f(X)$ (and, moreover, the image under $f$ of any constructible subset in $X$) is a constructible subset in $Y$. This is related to the fact that "algebraic" conditions determine the constructible subsets of an algebraic variety.
A mapping $h:X\rightarrow T$ is called constructible if $h(X)$ is finite and if for any point $t\in T$ the pre-image $h^{-1}(t)$ is a constructible subset in $X$.
References
| [1] | A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique" , I. Le langage des schémas , Springer (1971) MR0217085 Zbl 0203.23301 |
| [2] | A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201 |
How to Cite This Entry:
Constructible subset. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Constructible_subset&oldid=15045
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