On the relative K-group in the ETNC

TL;DR

This paper demonstrates that the relative K-group in the Burns-Flach formulation of the ETNC is equivalent to an ordinary K-group of locally compact modules, providing a new interpretation involving equivariant Haar measures.

Contribution

It establishes an isomorphism between the relative K-group in ETNC and an ordinary K-group, offering a novel perspective on the Tamagawa number as an equivariant Haar measure.

Findings

Relative K-group agrees with an ordinary K-group

Virtual objects correspond to equivariant Haar measures

Results hold for regular orders like hereditary orders

Abstract

We consider the Burns-Flach formulation of the equivariant Tamagawa number conjecture (ETNC). In their setup, a Tamagawa number is an element of a relative K-group. We show that this relative group agrees with an ordinary K-group, namely of the category of locally compact topological modules over the order. Its virtual objects are an equivariant Haar measure in a precise sense. We expect that all relative K-groups in the ETNC will have analoguous interpretations. At present, we need to restrict to regular orders, e.g. hereditary.