A family of totally ordered groups with some special properties

Olivos, Elena

Annales mathématiques Blaise Pascal, Tome 12 (2005) no. 1, pp. 79-90.

Let $K$ be a field with a Krull valuation $\left|\right|$ and value group $G\ne \left\{1\right\}$, and let $B_K$ be the valuation ring. Theories about spaces of countable type and Hilbert-like spaces in [1] and spaces of continuous linear operators in [2] require that all absolutely convex subsets of the base field $K$ should be countably generated as $B_K$-modules.

By [1] Prop. 1.4.1, the field $K$ is metrizable if and only if the value group $G$ has a cofinal sequence. We prove that for any fixed cardinality ${\aleph }_{\kappa }$, there exists a metrizable field $K$ whose value group has cardinality ${\aleph }_{\kappa }$. The existence of a cofinal sequence only depends on the choice of some appropriate ordinal $\alpha$ which has cardinality ${\aleph }_{\kappa }$ and which has cofinality $\omega$.

By [2] Prop. 1.4.4, the condition that any absolutely convex subset of $K$ be countably generated as a $B_K$-module is equivalent to the fact that the value group has a cofinal sequence and each element in the completion $G^{\#}$ is obtained as the supremum of a sequence of elements of $G$. We prove that for any fixed uncountable cardinal ${\aleph }_{\kappa }$ there exists a metrizable field $K$ of cardinality ${\aleph }_{\kappa }$ which has an absolutely convex subset that is not countably generated as a $B_K$-module.

We prove also that for any cardinality ${\aleph }_{\kappa }>{\aleph }_0$ for the value group the two conditions (the whole group has a cofinal sequence and every subset of the group which is bounded above has a cofinal sequence) are logically independent.

[1] H. Ochsenius, W. Schikhof Banach spaces over fields with an infinite rank valuation, In p-Adic Functional Analysis, Lecture Notes in pure and applied mathematics 207, edited by J. Kakol, N. De Grande-De Kimpe and C. Pérez García. Marcel Dekker (1999), pp. 233-293 | MR | Zbl

[2] H. Ochsenius, W. Schikhof Lipschitz operators in Banach spaces over Krull valued fields, Report N. 0310, University of Nijmegen, The Netherlands, Volume 13 (2003) | Zbl

[3] Jech, T. Set Theory, San Diego Academic Press, USA, 1978 | MR | Zbl

[4] Ribenboim, P. Théorie des valuations, Les Presses de l’Université de Montréal, Montréal, Canada, 1968 | Zbl

[5] Ribenboim, P. The theory of classical valuations, Springer-Verlag, 1998 | MR | Zbl