Constructible topology

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

With respect to this topology, 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 is a von Neumann regular ring, where 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

References

  1. Some authors prefer the term quasicompact here.
  2. "Lemma 5.23.8 (0905)—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-09-20.
  3. "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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