Adelic descent for K-theory

TL;DR

This paper establishes an adelic descent equivalence for localizing invariants like algebraic K-theory on Noetherian schemes, linking global invariants to limits over Beilinson's reduced adeles.

Contribution

It proves a new adelic descent theorem for localizing invariants, connecting algebraic K-theory to Beilinson's semi-cosimplicial rings of reduced adeles.

Findings

Proves an equivalence between E(X) and a limit over adelic complexes.

Establishes cubical descent via exact sequences of perfect module categories.

Applies to invariants such as algebraic K-theory of Bass-Thomason.

Abstract

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 $E(X)\simeq \lim E(A^{\cdot}_{\text{red}}(X))$, where $A^{\cdot}_{\text{red}}(X)$ 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.