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

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

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