A homological approach to chromatic complexity of algebraic K-theory

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.

36 pages. Significantly revised after referee feedback. Accepted for publication at JPAA

References in corpus (3)

• Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
• The homotopy groups of the algebraic K-theory of the sphere spectrum
• Commuting unbounded homotopy limits with Morava K-theory

Cited by in corpus (1)

• Commuting unbounded homotopy limits with Morava K-theory