Profinite Rigidity over Noetherian Domains

TL;DR

This paper explores how modules over Noetherian domains can be uniquely determined by their finite images, establishing foundational results and demonstrating profinite rigidity in various algebraic contexts.

Contribution

It introduces the concept of profinite rigidity for modules over Noetherian domains, providing characterizations and conditions under which modules are uniquely determined by their finite quotients.

Findings

Free modules are profinitely rigid under certain homological conditions.

All finitely generated modules over a PID are profinitely rigid.

Solvable Baumslag–Solitar groups are shown to be profinitely rigid.

Abstract

We initiate the study of profinite rigidity for modules over a Noetherian domain: to what extent are these objects determined by their finite images? We establish foundational statements in analogy to classical results in the category of groups. We describe three profinite invariants of modules over any Noetherian domain $\Lambda$. We show that free modules are profinitely rigid when $\Lambda$ satisfies a homological condition, and characterise the profinite genus of all modules when $\Lambda$ is a Dedekind domain. In the case where $\Lambda$ is a PID, we find that all finitely generated modules are profinitely rigid. As an application, we prove that solvable Baumslag--Solitar groups are profinitely rigid in the absolute sense. These are the first examples of absolute profinite rigidity among non-abelian one-relator groups and among non-LERF groups.