New York Journal of Mathematics
Volume 29 (2023), 1-28


Hyungseop Kim

Adelic descent for K-theory

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Published: December 17, 2022.
Keywords: Algebraic K-theory, localizing invariants, descent for K-theory, higher adeles.
Subject [2010]: 19E08, 19D99

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 between E(X) and
lim E(A.red(X)), where A.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.


The author would like to thank his advisor Michael Groechenig for his suggestion of the problem and support through numerous discussions, without which this project would not have led anywhere. I also would like to thank Benjamin Antieau for helpful discussions through email, Oliver Braunling for pointing out Balmer's paper [Bal08], and the anonymous referee for careful reading and helpful comments. The author was supported by the CMK Foundation.

Author information

Hyungseop Kim:
Department of Mathematics
University of Toronto
Bahen Centre, 40 St. George St.
Toronto, Ontario M5S 2E4, Canada