K-Theory of Locally Compact Modules over Rings of Integers

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 that might be of independent interest. As in Clausen's work, our computation works for all localizing invariants, not just K-theory.