Prüfer intersection of valuation domains of a field of rational functions

Abstract

Let V be a rank one valuation domain with quotient field K. We characterize the subsets S of V for which the ring of integer-valued polynomials Int(S,The characterization is obtained by means of the notion of pseudomonotone sequence and pseudo-limit in the sense of Chabert, which generalize the classical notions of pseudo-convergent sequence and pseudo-limit by Ostrowski and Kaplansky, respectively. We show that Int(S, V ) is Prüfer if and only if no element of the algebraic closure K of K is a pseudo-limit of a pseudo-monotone sequence contained in S, with respect to some extension of V to K. This result expands a recent result by Loper and Werner.

The Zariski–Riemann space of valuation domains associated to pseudo-convergent sequences

“…The three classes of pseudo-convergent, pseudo-divergent and pseudo-stationary sequences have been introduced in [6] and together form the class of pseudo-monotone sequences [17]. Most of the notions introduced for pseudo-convergent sequences, like the breadth and the valuation domain V E , can be generalized to pseudo-monotone sequences, see [20].…”

“…In Section 5, we look at the same partitions, but on the sets V div and V staz of extensions induced, respectively, by pseudo-divergent and pseudo-stationary sequences (the other type of pseudo-monotone sequences beyond the pseudo-convergent ones, see [6,17,20]). Using a quotient onto the space Zar(k(t)|k) (where k is the residue field of V ) we first show that Zar(K(X)|V ) cons is not metrizable if k is uncountable (Proposition 5.3); then, with a similar method, we show that V div (•, δ) is not Hausdorff (with respect to the Zariski topology) when δ belongs to the value group of V (Proposition 5.4).…”

The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences

“…(c) Let E = {s n } n∈N be of algebraic type with finite breadth, and let {δ n } n∈N be the gauge of E. Since V E ⊆ W E and W E has rank 1, conditions (iii) and (iv) are clearly equivalent. Since by Lemma 4.8 w E = v α,δ , by [18,Lemma 3.5] we have that (i) is equivalent to (ii). Clearly, (iv) implies (v).…”

“…VI, §. 10, Lemme 1], and v α,δ is residually transcendental over v if and only if δ has finite order over Γ v [18,Lemma 3.5]. Furthermore, every residually transcendental extension of V can be written as W ∩ K(X), where W is a valuation domain of K(X) associated to a monomial valuation [1,2].…”

Integer-valued polynomials, Prüfer domains and the stacked bases property

“…Theorem 2.1. [24] The ring Int(E, V ) is Prüfer if and only if the only pseudomonotone sequences contained in E are either Cauchy sequences or pseudo-convergent sequences of transcendental type.…”