Constructible topology

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

• Constructible set (topology)

• 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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