$K$-theory of locally compact modules over orders

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.