Editing Spectrum of a ring (section)

== Non-Zariski topologies on a prime spectrum ==
{{expand section|date=June 2020}}

Some authors (notably M. Hochster) consider topologies on prime spectra other than the Zariski topology.

First, there is the notion of [[constructible topology]]: given a ring ''A'', the subsets of <math>\operatorname{Spec}(A)</math> of the form <math>\varphi^*(\operatorname{Spec} B), \varphi: A \to B</math> satisfy the axioms for closed sets in a topological space. This topology on <math>\operatorname{Spec}(A)</math> is called the constructible topology.{{sfnp|Atiyah|Macdonald|1969|loc=Ch. 5, Exercise 27|ps=}}{{sfnp|Tarizadeh|2019|ps=}}<!-- (cf. [[faithfully flat ring homomorphism]])-->

In {{harvp|Hochster|1969}}, Hochster considers what he calls the patch topology on a prime spectrum.{{sfnp|Kock|2007|ps=}}{{sfnp|Fontana|Loper|2008|ps=}}{{sfnp|Brandal|1979|ps=}} By definition, the patch topology is the smallest topology in which the sets of the forms <math>V(I)</math> and <math>\operatorname{Spec}(A) - V(f)</math> are closed.