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 (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 ${\displaystyle A/\operatorname {nil} (A)}$ is a von Neumann regular ring, where ${\displaystyle \operatorname {nil} (A)}$ is the nilradical of A.^[2]

Despite the terminology being similar, the constructible topology is not the same as the set of all constructible sets.^[3]

See also

• Constructible set (topology)

References

1. Some authors prefer the term quasicompact here.
2. "Lemma 5.23.8 (0905)—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-09-20.
3. "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