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arXiv · 2011.08233

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

Dustin Clausen·Akhil Mathew·Niko Naumann·Justin Noel
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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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Dustin Clausen, Akhil Mathew, Niko Naumann, Justin Noel. 2020-11-16. Descent and vanishing in chromatic algebraic $K$-theory via group actions. https://arxiv.org/abs/2011.08233

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