$K$-theory of locally compact modules over orders

TL;DR

This paper introduces a rapid method for computing the K-theory of locally compact modules over orders in semisimple rational algebras by using quotient categories and torsion theory.

Contribution

It provides a novel approach leveraging localization and torsion theory to simplify K-theory calculations for these modules.

Findings

Effective computation of K-theory for locally compact modules over orders.

Utilization of quotient categories to facilitate calculations.

Application of recent localization formalism to exact categories.

Abstract

We present a quick approach to computing the $K$-theory of the category of locally compact modules over any order in a semisimple $\mathbb{Q}$-algebra. We obtain the $K$-theory by first quotienting out the compact modules and subsequently the vector modules. Our proof exploits the fact that the pair (vector modules plus compact modules, discrete modules) becomes a torsion theory after we quotient out the finite modules. Treating these quotients as exact categories is possible due to a recent localization formalism.