Algebraic $K$-theory of $\text{THH}({\mathbb{F}}_p)$

Abstract

In this work we study the $E_{\infty }$-ring $\text{THH}({\mathbb{F}}_p)$ as a graded spectrum. Following an identification at the level of $E_2$-algebras with ${\mathbb{F}}_p[\Omega S^3]$, the group ring of the $E_1$-group $\Omega S^3$ over ${\mathbb{F}}_p$, we show that the grading on $\text{THH}({\mathbb{F}}_p)$ arises from decomposition on the cyclic bar construction of the pointed monoid $\Omega S^3$. This allows us to use trace methods to compute the algebraic $K$-theory of $\text{THH}({\mathbb{F}}_p)$. We also show that as an $E_2$ $H{\mathbb{F}}_p$-ring, $\text{THH}({\mathbb{F}}_p)$ is uniquely determined by its homotopy groups. These results hold in fact for $\text{THH}(k)$, where $k$ is any perfect field of characteristic $p$. Along the way we expand on some of the methods used by Hesselholt-Madsen and later by Speirs to develop certain tools to study the THH of graded ring spectra and the algebraic $K$-theory of formal DGAs.

References in corpus (3)

• Higher Topos Theory
• The geometry of filtrations
• On The Algebraic $K$-Theory of Double Points