The Zariski–Riemann space 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 study 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 .

Topological properties of subsets of the Zariski space

“…To each pseudo-convergent sequence E we associate the map w E : K(X) −→ R ∪ {∞} such that [17,Propositions 4.3 and 4.4] w E (φ) := lim…”

“…If w E is a valuation, the corresponding valuation ring W E is a one-dimensional extension of V to K(X); if K is algebraically closed, then every rank-one extension of V to K(X) is in this form [15,16]. We denote the set of all rings in the form W E as W: then, the Zariski and the constructible topologies agree on W, and under them W is a regular zero-dimensional space that is not compact [17,Propositions 6.3 and 6.4].…”

“…The valuation v E can also be described explicitly as a map into R 2 (see [17,Theorem 4.10]). We denote the set of all the V E as V: then, V is a regular space in both the Zariski and the constructible topologies [17,Theorem 6.15], but the two topologies agree on V if and only if the residue field of V is finite [17,Proposition 6.11]. There is also a map…”

“…that, under the Zariski topology, is continuous and injective, but not a topological embedding [17,Proposition 6.13].…”

“…Then, the Zariski and the constructible topologies agree on V(•, δ) [18,Theorem 3.5]; furthermore, this topology is also generated by the ultrametric distance…”