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arXiv · 1705.01301

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

Dario Spirito
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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.

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Dario Spirito. 2017-05-03. Non-compact subsets of the Zariski space of an integral domain. https://arxiv.org/abs/1705.01301

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