# Non-compact subsets of the Zariski space of an integral domain

## TL;DR

This paper investigates the topological properties of the Zariski space of valuation overrings of an integral domain, showing that certain subsets are not compact and that the space often lacks Noetherianity, with applications to Kronecker function rings.

## Contribution

It provides a new criterion for non-compactness in the Zariski space and demonstrates that this space is frequently non-Noetherian, advancing understanding of its topological structure.

## Key findings

- Sets of valuation overrings excluding a minimal valuation are not compact.
- The Zariski space of an integral domain is often non-Noetherian.
- Applications to Kronecker function rings and Noetherian overrings.

## Abstract

Let $V$ be a minimal valuation overring of an integral domain $D$ and let $\mathrm{Zar}(D)$ be the Zariski space of the valuation overrings of $D$. Starting from a result in the theory of semistar operations, we prove a criterion under which the set $\mathrm{Zar}(D)\setminus\{V\}$ is not compact. We then use it to prove that, in many cases, $\mathrm{Zar}(D)$ is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings.