Constructible topology

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

With respect to this topology, $\mathrm{Spec}(A)$ 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 $A/\mathrm{nil}(A)$ is a von Neumann regular ring, where $\mathrm{nil}(A)$ 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

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