Topological properties of localizations, flat overrings and sublocalizations

Abstract

We study the set of localizations of an integral domain from a topological point of view, showing that it is always a spectral space and characterizing when it is a proconstructible subspace of the space of all overrings. We then study the same problems in the case of quotient rings, flat overrings and sublocalizations.

The Derived Sequence of a Pre-Jaffard Family

“…Every flat overring is a sublocalization [10, Corollary to Theorem 2], but the converse is not true (see [11] and [12,Example 6.3]). We can characterize when a sublocalization is flat.…”

The derived sequence of a pre-Jaffard family

“…Every flat overring is a sublocalization [14, Corollary to Theorem 2], but the converse is not true (see [8] and [16,Example 6.3]). We can characterize when a sublocalization is flat.…”

Isolated points of the Zariski space

“…Both of them give rise to spectral spaces (in particular, they are compact); furthermore, the constructible topology gains the property of being Hausdorff, and plays an important role in the topological characterization of spectral spaces (see for example Hochster's article [14]). The constructible topology can also be studied through ultrafilters [7], and this point of view allows to give many examples of spectral spaces, for example by finding them inside other spectral spaces (see [22,Example 2.2(1)] for some very general constructions, [29] for examples in the overring case, and [10,9] for examples in the semistar operations setting).…”