p-adic functionals on finite-rank torsion-free abelian groups

TL;DR

This paper studies p-adic functionals on finite-rank torsion-free abelian groups, describing their structure via inductive limits and showing invariance properties under quasi-isomorphism.

Contribution

It provides a detailed description of p-adic functionals on finite-rank torsion-free abelian groups using inductive sequences, especially in stationary cases, extending previous classifications.

Findings

Characterization of p-adic functionals via inductive limits

Stationary inductive sequences lead to simple module descriptions

Class of stationary limits is closed under quasi-isomorphism

Abstract

Let p be a prime and G be a torsion-free abelian group. A homomorphism from G to the p-adic integers is called a p-adic functional on G. If G has finite rank, then G can be represented as an inductive limit of an inductive sequence of free abelian groups of the same rank, and the group of all p-adic functionals on G is described in terms of this inductive sequence. If this inductive sequence is stationary--i.e., if the homomorphism is the same at every stage--then the group of p-adic functionals is described in particularly simple terms, as a right-submodule that is invariant under the module homomorphism that this group homomorphism induces. It is shown that the class consisting of all such stationary inductive limits is closed under quasi-isomorphism; this strengthens a previous classification result of Dugas.