Equivariant Structure on Smash Powers

TL;DR

This paper develops a categorical framework for equivariant structures on smash powers of commutative orthogonal ring spectra, generalizing topological Hochschild homology and related constructions for applications in algebraic K-theory.

Contribution

It introduces a new categorical approach to equivariant smash powers, extending existing theories to more general groups and spaces, with applications to algebraic K-theory.

Findings

Framework for equivariant smash powers of ring spectra

Generalization of cyclotomic structures in a categorical setting

Application to iterated algebraic K-theory

Abstract

We provide foundations for dealing with the equivariant structure of "smash powers" of commutative orthogonal ring spectra. The category of commutative orthogonal ring spectra $A$ is tensored over spaces $X$, so that $A \otimes X$ is a commutative orthogonal ring spectrum. If $X$ is a discrete space, this is literally the smash power of $A$ with itself indexed over $X$, and we keep this language also in the nondiscrete case. In particular $A \otimes S^1$ is a model for topological Hochschild homology. We provide a framework where a generalization of the cyclotomic structure of topological Hochschild homology is visible in a categorical framework, also for more general $G$ and $X$. Similar situations have been studied by others, e.g., in Hill, Hopkins and Ravenel's treatment of the norm construction and Brun, Carlsson, Dundas' covering homology. In the case of non-commutative $A$ and $X=S^1$, the situation is somewhat easier and has already been covered by Kro. We are motivated by applications to $G$ being a torus in order to study the iterated algebraic $K$-theory, and have to develop a categorical theory that in some ways goes beyond what has been done before. Most of the material appeared in the last author's thesis which was defended in 2011. We apologize for the delay.