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.

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

References

↑ Some authors prefer the term quasicompact here.
↑ "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

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Constructible topology

See also

References