Descent and vanishing in chromatic algebraic \(K\)-theory via group actions

We prove some \(K\)-theoretic descent results for finite group actions on stable \(\infty\)-categories, including the \(p\)-group case of the Galois descent conjecture of Ausoni-Rognes. We also prove vanishing results in accordance with Ausoni-Rognes's redshift philosophy: in particular, we sho...

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Bibliographic Details
Published in:arXiv.org 2022-11
Main Authors: Clausen, Dustin, Mathew, Akhil, Naumann, Niko, Noel, Justin
Format: Article
Language:English
Subjects:
Descent
Red shift
Online Access:Get full text
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Summary:We prove some \(K\)-theoretic descent results for finite group actions on stable \(\infty\)-categories, including the \(p\)-group case of the Galois descent conjecture of Ausoni-Rognes. We also prove vanishing results in accordance with Ausoni-Rognes's redshift philosophy: in particular, we show that if \(R\) is an \(\mathbb{E}_\infty\)-ring spectrum with \(L_{T(n)}R=0\), then \(L_{T(n+1)}K(R)=0\). Our key observation is that descent and vanishing are logically interrelated, permitting to establish them simultaneously by induction on the height.
ISSN:2331-8422