Constructible topology

In commutative algebra, the constructible topology on the spectrum [math]\displaystyle{ \operatorname{Spec}(A) }[/math] of a commutative ring [math]\displaystyle{ A }[/math] is a topology where each closed set is the image of [math]\displaystyle{ \operatorname{Spec} (B) }[/math] in [math]\displaystyle{ \operatorname{Spec}(A) }[/math] for some algebra B over A. An important feature of this construction is that the map [math]\displaystyle{ \operatorname{Spec}(B) \to \operatorname{Spec}(A) }[/math] is a closed map with respect to the constructible topology. With respect to this topology, [math]\displaystyle{ \operatorname{Spec}(A) }[/math] is a compact, Hausdorff, and totally disconnected topological space (i.e., a Stone space). In general, the constructible topology is a finer topology than the Zariski topology, and the two topologies coincide if and only if [math]\displaystyle{ A / \operatorname{nil}(A) }[/math] is a von Neumann regular ring, where [math]\displaystyle{ \operatorname{nil}(A) }[/math] is the nilradical of A.

Despite the terminology being similar, the constructible topology is not the same as the set of all constructible sets.

See also

• Constructible set (topology)

References

• Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, p. 87, ISBN 978-0-201-40751-8 
• Knight, J. T. (1971), Commutative Algebra, Cambridge University Press, pp. 121–123, ISBN 0-521-08193-9