Adelic descent for K-theory

We prove an adelic descent result for localizing invariants: for each Noetherian scheme $X$ of finite Krull dimension and any localizing invariant $E$, e.g., algebraic K-theory of Bass-Thomason, there is an equivalence $E(X)\simeq \lim E(A_{\text{red}}^{\cdot }(X))$, where $A_{\text{red}}^{\cdot }(X)$ denotes Beilinson’s semi-cosimplicial ring of reduced adeles on $X$. We deduce the equivalence from a closely related cubical descent result, which we prove by establishing certain exact sequences of perfect module categories over adele rings.