Factor equivalence of Galois modules and regulator constants

TL;DR

This paper explores the relationship between factor equivalence and regulator constants in Galois module theory, revealing their close connection and deriving new results on higher K-groups of rings of integers.

Contribution

It demonstrates the equivalence of two major approaches in Galois module study and applies this to obtain new factorisability results for higher K-groups.

Findings

Factor equivalence and regulator constants are closely related.

Results from both approaches can be interpreted interchangeably.

Derived a factorisability theorem for higher K-groups of rings of integers.

Abstract

We compare two approaches to the study of Galois module structures: on the one hand factor equivalence, a technique that has been used by Fr\"ohlich and others to investigate the Galois module structure of rings of integers of number fields and of their unit groups, and on the other hand regulator constants, a set of invariants attached to integral group representations by Dokchitser and Dokchitser, and used by the author, among others, to study Galois module structures. We show that the two approaches are in fact closely related, and interpret results arising from these two approaches in terms of each other. We also use this comparison to derive a factorisability result on higher $K$-groups of rings of integers, which is a direct analogue of a theorem of de Smit on $S$-units.