Constructible topology

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

## References

1. ↑ Some authors prefer the term quasicompact here.
2. ↑ "Lemma 5.23.8 (0905)—The Stacks project". https://stacks.math.columbia.edu/tag/0905.
3. ↑ "Reconciling two different definitions of constructible sets". https://math.stackexchange.com/q/1964351.

* 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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