Constructible topology

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

With respect to this topology, ${\displaystyle \operatorname {Spec} (A)}$ 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 ${\displaystyle A/\operatorname {nil} (A)}$ is a von Neumann regular ring, where ${\displaystyle \operatorname {nil} (A)}$ is the nilradical of A.

Despite the terminology being similar, the constructible topology is not the same as the set of all constructible sets.[2]

See also

• Constructible set (topology)

References

1. Some authors prefer the term quasicompact here.
2. "Reconciling two different definitions of constructible sets". math.stackexchange.com. Retrieved 2016-10-13.

• 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