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

Abstract

Let $V$ be a minimal valuation overring of an integral domain $D$ and let $\operatorname{Zar}(D)$ 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 $\operatorname{Zar}(D)\setminus\{V\}$ is not compact. We then use it to prove that, in many cases, $\operatorname{Zar}(D)$ is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings.