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Journal of the European Mathematical Society

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Volume 22, Issue 4, 2020, pp. 1149–1200
DOI: 10.4171/JEMS/942

Published online: 2020-01-03

Descent in algebraic $K$-theory and a conjecture of Ausoni–Rognes

Dustin Clausen[1], Akhil Mathew[2], Niko Naumann[3] and Justin Noel[4]

(1) University of Copenhagen, Denmark
(2) University of Chicago, USA
(3) University of Regensburg, Germany
(4) University of Regensburg, Germany

Let $A \to B$ be a $G$-Galois extension of rings, or more generally of $\mathbb{E}_\infty$-ring spectra in the sense of Rognes. A basic question in algebraic $K$-theory asks how close the map $K(A) \to K(B)^{hG}$ is to being an equivalence, i.e., how close algebraic $K$-theory is to satisfying Galois descent. An elementary argument with the transfer shows that this equivalence is true rationally in most cases of interest. Motivated by the classical descent theorem of Thomason, one also expects such a result after periodic localization.

We formulate and prove a general result which enables one to promote rational descent statements as above into descent statements after periodic localization. This reduces the localized descent problem to establishing an elementary condition on $K_0(-)\otimes \mathbb{Q}$. As applications, we prove various descent results in the periodically localized $K$-theory, $TC$, $THH$, etc. of structured ring spectra, and verify several cases of a conjecture of Ausoni and Rognes.

Keywords: Algebraic $K$-theory, Galois descent, noncommutative motives, nilpotence

Clausen Dustin, Mathew Akhil, Naumann Niko, Noel Justin: Descent in algebraic $K$-theory and a conjecture of Ausoni–Rognes. J. Eur. Math. Soc. 22 (2020), 1149-1200. doi: 10.4171/JEMS/942