A homological approach to chromatic complexity of algebraic K-theory

Abstract

The family of Thom spectra $y\left(n\right)$ 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 $y\left(n\right)$ 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 peridic 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.