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Difference between revisions of "Constructible subset"

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''of an algebraic variety''
 
''of an algebraic variety''
  
A finite union of locally closed (in the [[Zariski topology|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253001.png" /> is a morphism of algebraic varieties, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253002.png" /> (and, moreover, the image of any constructible subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253003.png" />) is a constructible subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253004.png" />. This is related to the fact that  "algebraic"  conditions determine the constructible subsets of an algebraic variety.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253005.png" /> is called constructible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253006.png" /> is finite and if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253007.png" /> the pre-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253008.png" /> is a constructible subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025300/c0253009.png" />.
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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,   J. Dieudonné,   "Eléments de géometrie algébrique" , '''I. Le langage des schémes''' , 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>
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<table>
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<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>
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<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>
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</table>
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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=21857
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article