Valuations centered at a two-dimensional regular local domain: infima and topologies

Abstract

Take a two-dimensional regular local domain $R$. The space of all valuations centered at $R$ has a non-metric tree structure, called the valuative tree of $R$. However, the notion of non-metric tree appearing in the literature does not guarantee the existence of infimum for a non-empty set of valuations. We give a more general definition of a rooted non-metric tree and prove that the valuative tree has this more general property. We also generalize some topological results related to a non-metric tree. For instance, we show that the weak tree topology is always coarser than the metric topology given by any parametrization.