Topological properties of subsets of the Zariski space

Volume 121 / 2020

Dario Spirito Banach Center Publications 121 (2020), 161-167 MSC: Primary 13F30; Secondary 12J20, 13A15, 13A18, 54D30, 54E35. DOI: 10.4064/bc121-15

Abstract

We study the properties of some distinguished subspaces of the Zariski space Zar$(K|D)$ of a field $F$ over a domain $D$, in particular the topological properties of subspaces defined through algebraic means. We are mainly interested in two classes of problems: understanding when spaces of the form Zar$(K|D)\setminus \{V\}$ are compact (which is strongly linked to the problem of determining when Zar$(K|D)$ is a Noetherian space), and studying spaces of rings defined through pseudo-convergent sequences on an (arbitrary, but fixed) rank one valuation domain.

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