Topology on rational points over higher local fields

TL;DR

This paper develops a topology on the set of rational points over higher local fields, extending Weil's constructions, and studies its properties in the context of higher dimensional number theory and algebraic geometry.

Contribution

It introduces a new sequential topology on rational points over higher local fields, generalizing classical constructions to higher dimensions.

Findings

Established a topology on rational points over higher local fields.

Proved functoriality of the topology.

Derived properties of the topological structure.

Abstract

We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields and their rings of integers and include higher local fields. Our results extend the constructions of Weil over (one-dimensional) local fields. We establish the existence of an appropriate topology on the set of rational points of schemes of finite type over any of the rings considered, study the functoriality of this construction and deduce several properties.