K-theory of locally compact modules over rings of integers

TL;DR

This paper extends the computation of K-theory for locally compact modules over rings of integers in number fields, using a novel approach based on calculus of fractions, applicable to all localizing invariants.

Contribution

It introduces a new proof method for K-theory computations of locally compact modules over rings of integers, differing from previous homotopy coherent cone constructions.

Findings

Computed K-theory for locally compact modules over rings of integers

Established concrete exact category models for certain quotients

Method applies to all localizing invariants, not just K-theory

Abstract

We generalize a recent result of Clausen: For a number field with integers O, we compute the K-theory of locally compact O-modules. For the rational integers this recovers Clausen's result as a special case. Our method of proof is quite different: Instead of a homotopy coherent cone construction in infinity categories, we rely on calculus of fraction type results in the style of Schlichting. This produces concrete exact category models for certain quotients, a fact which might be of independent interest. As in Clausen's work, our computation works for all localizing invariants, not just K-theory.