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Descent and vanishing in chromatic algebraic $K$-theory via group actions

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arxiv 2011.08233 v2 pith:NMMIJK34 submitted 2020-11-16 math.KT math.AT

Dustin Clausen , Akhil Mathew , Niko Naumann , Justin Noel This is my paper · ORCID
classification math.KTmath.AT
keywords descentgroupvanishingactionsausoni-rognesinftyproveresults
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abstract

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.

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