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

TL;DR

This paper investigates Galois descent in algebraic $K$-theory and related invariants for ring spectra, providing a general method to upgrade rational descent results to periodic localizations and confirming cases of a conjecture by Ausoni and Rognes.

Contribution

It formulates a general theorem that promotes rational descent to periodic localization, simplifying the descent problem to an elementary condition on $K_0$ and verifying parts of the Ausoni-Rognes conjecture.

Findings

Established descent results in periodic localized $K$-theory, $TC$, and $THH$.

Reduced localized descent to a condition on $K_0(-) ensor Q$.

Verified several cases of the Ausoni-Rognes conjecture.

Abstract

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 periodic localized $K$-theory, $TC$, $THH$, etc. of structured ring spectra, and verify several cases of a conjecture of Ausoni and Rognes.