Constructible topology

In commutative algebra, the constructible topology on the spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ is a topology where each closed set is the image of $\operatorname{Spec} (B)$ in $\operatorname{Spec}(A)$ for some algebra B over A. An important feature of this construction is that the map $\operatorname{Spec}(B) \to \operatorname{Spec}(A)$ is a closed map with respect to the constructible topology.

With respect to this topology, $\operatorname{Spec}(A)$ is a compact,^[1] Hausdorff, and totally disconnected topological space. In general the constructible topology is a finer topology than the Zariski topology, but the two topologies will coincide if and only if $A / \operatorname{nil}(A)$ is a von Neumann regular ring, where $\operatorname{nil}(A)\,$ is the nilradical of A.

See also

• Constructible set (topology)

References

1. Some authors prefer the term quasicompact here.

• 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