Metrizability of spaces of valuation domains associated to pseudo-convergent sequences

Abstract

Let $V$ be a valuation domain of rank one with quotient field $K$. We study the set of extensions of $V$ to the field of rational functions $K(X)$ induced by pseudo-convergent sequences of $K$ from a topological point of view, endowing this set either with the Zariski or with the constructible topology. In particular, we consider the two subspaces induced by sequences with a prescribed breadth or with a prescribed pseudo-limit. We give some necessary conditions for the Zariski space to be metrizable (under the constructible topology) in terms of the value group and the residue field of $V$.

arXiv commentpp. 1-22, final version, to appear in J. Algebra Appl. (2021). arXiv admin note: substantial text overlap with arXiv:1809.09539

References in corpus (4)

• Extending valuations to the field of rational functions using pseudo-monotone sequences
• Prüfer intersection of valuation domains of a field of rational functions
• Transcendental extensions of a valuation domain of rank one
• The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences