Boundary valuation domains

Filtered Modules over Discrete Valuation Domains

We consider a uni ed setting for studying local valuated groups and coset-valuated groups, emphasizing the associated ltrations rather than the values of elements. Stable exact sequences, projectives and injectives are identi ed in the encompassing category, and in the category corresponding to coset-valuated groups.

Divisible Uniserial Modules over Valuation Domains

Sigma-cotorsion Modules over Valuation Domains

We give a characterization of Σ-cotorsion modules over valuation domains in terms of descending chain conditions on certain chains of definable subgroups. We prove that pure submodules, direct products and modules elementarily equivalent to a Σ-cotorsion module are again Σ-cotorsion. Moreover, we describe the structure of Σ-cotorsion modules.

Intersections of valuation overrings of two-dimensional Noetherian domains

We survey and extend recent work on integrally closed overrings of two-dimensional Noetherian domains, where such overrings are viewed as intersections of valuation overrings. Of particular interest are the cases where the domain can be represented uniquely by an irredundant intersection of valuation rings, and when the valuation rings can be chosen from a Noetherian subspace of the Zariski-Rie...

A Classical Result on Maximal Valuation Domains Revisited

We prove that a non linearly compact valuation domain R admits a proper immediate extension S. This is the main point of Kaplansky’s classical result that a valuation domain is linearly compact if and only if it is maximal. In fact, Kaplansky’s original proof, as well as later versions of it, do not show that R and S have the same residue field.