A homological approach to chromatic complexity of algebraic K-theory

Gabriel Angelini-Knoll; J. D. Quigley

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arXiv · 1908.09164

A homological approach to chromatic complexity of algebraic K-theory

Gabriel Angelini-Knoll·J. D. Quigley

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Abstract

The family of Thom spectra $y(n)$ interpolates between the sphere spectrum and the mod two Eilenberg--MacLane spectrum. Computations of Mahowald, Ravenel, Shick, and the authors show that the associative ring spectrum $y(n)$ has type $n$. Using trace methods, we give evidence that algebraic K-theory preserves this chromatic complexity. Our approach sheds light on the chromatic complexity of topological negative cyclic homology and topological periodic cyclic homology, which approximate algebraic K-theory and are of independent interest. Our main contribution is a homological approach that can be applied in great generality, such as to associative ring spectra $R$ without additional structure whose coefficient rings are not completely understood.

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BibTeX RIS

Gabriel Angelini-Knoll, J. D. Quigley. 2019-08-24. A homological approach to chromatic complexity of algebraic K-theory. https://arxiv.org/abs/1908.09164

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