Constructible topology explained

\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 (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.^[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)

Notes and References

1. Some authors prefer the term quasicompact here.
2. Web site: Lemma 5.23.8 (0905)—The Stacks project . 2022-09-20 . stacks.math.columbia.edu.
3. Web site: Reconciling two different definitions of constructible sets. math.stackexchange.com. 2016-10-13.