The $R(S^1)$–graded equivariant homotopy of THH($\mathbb{F}_p$)

Abstract

The main result of this paper is the computation of TR${}_{\alpha }^n\left({\mathbb{F}}_p;p\right)$ for $\alpha \in R\left(S^1\right)$. These $R\left(S^1\right)$–graded TR–groups are the equivariant homotopy groups naturally associated to the $S^1$–spectrum THH$\left({\mathbb{F}}_p\right)$, the topological Hochschild $S^1$–spectrum. This computation, which extends a partial result of Hesselholt and Madsen, provides the first example of the $R\left(S^1\right)$–graded TR–groups of a ring. These groups arise in algebraic $K$–theory computations and are particularly important to the understanding of the algebraic $K$–theory of non-regular schemes.