Topological properties of sets and convergence rings of a multidimensional complete field

Abstract

This article continues the series of works by the first author on the convergence of sequences and series in multidimensional local and complete fields.
Multidimensional fields are a chain of discretely normalized fields, where each subsequent field is the residue field of the previous one. As a result, the elements are represented as a
series, and when using the standard topology of discrete normalization, the series defining the elements of the field may not necessarily converge. Therefore, on multidimensional complete fields, a complexly constructed Parshin topology is used, taking into account the topologies of the residue fields (see [15], [5], and [6]). In this topology, the series of all elements of the multidimensional field converge. However, another important property is not satisfied in the Parshin topology, which is the convergence of all power series with coefficients from the ring of integers when the element of the maximal ideal is substituted for the variable.
In [9] by the first author the concept of a convergence set is introduced, which is a set such that a series with coefficients from this set converges on a maximal ideal, and a criterion for a
convergence set is proved. In [10], convergence sets are studied using their multi-indices, which form a convergence monoid, and in [8], rings that are convergence sets are constructed and their
properties are studied. In this work, it is shown that the additive shift of a convergence set gives a convergence set, that any convergence set is sequentially closed, and that a convergent sequence always forms a convergence set. These statements provide a convenient sufficient condition for a sequence to
be infinitesimal and allow the construction of a convergence ring that contains the limit of a convergent sequence and all its members.