New distinguished classes of spectral spaces: a survey

TL;DR

This survey introduces new classes of spectral spaces arising in multiplicative ideal theory, extending the topology of semistar operations and exploring their properties and applications in algebraic structures.

Contribution

It presents novel classes of spectral spaces not directly linked to prime spectra of rings, using ultrafilter topology for characterization and extending the understanding of spectral spaces in algebra.

Findings

New classes of spectral spaces introduced

Ultrafilter topology characterizes spectral spaces

Applications in multiplicative ideal theory discussed

Abstract

In the present survey paper, we present several new classes of Hochster's spectral spaces "occurring in nature", actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The general setting is the space of the semistar operations (of finite type), endowed with a Zariski-like topology, which turns out to be a natural topological extension of the space of the overrings of an integral domain, endowed with a topology introduced by Zariski. One of the key tool is a recent characterization of spectral spaces, based on the ultrafilter topology, given in a paper by C. Finocchiaro in Comm. Algebra 2014. Several applications are also discussed.