  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.
