Spectral spaces of semistar operations

TL;DR

This paper explores the topological structure of various classes of semistar operations, establishing spectral space properties and homeomorphisms with other algebraic structures, thus advancing the understanding of their topological and algebraic interplay.

Contribution

It demonstrates that subspaces of finite type spectral semistar operations are spectral spaces and links stable semistar operations to Gabriel-Popescu localizing systems, extending known topological results.

Findings

Finite type spectral semistar operations form spectral spaces.

Stable semistar operations are homeomorphic to Gabriel-Popescu localizing systems.

Zariski topology on semistar operations matches the $b$-topology.

Abstract

We investigate, from a topological point of view, the classes of spectral semistar operations and of eab semistar operations, following methods recently introduced by Finocchiaro and Finocchiaro-Spirito in \cite{Fi, FiSp}. We show that, in both cases, the subspaces of finite type operations are spectral spaces in the sense of Hochster and, moreover, that there is a distinguished class of overrings strictly connected to each of the two types of collections of semistar operations. We also prove that the space of stable semistar operations is homeomorphic to the space of Gabriel-Popescu localizing systems, endowed with a Zariski-like topology, extending to the topological level a result established by Fontana-Huckaba in \cite{fohu}. As a side effect, we obtain that the space of localizing systems of finite type is also a spectral space. Finally, we show that the Zariski topology on the set of semistar operations is the same as the $b$-topology defined recently by B. Olberding \cite{ol, olb_noeth}.