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

Haldun Özgür Bayındır, Tasos Moulinos

Published 2020-09-12Version 1

In this work we study the $E_{\infty}$-ring $\text{THH}(\mathbb{F}_p)$ from various perspectives. Following an identification at the level of $E_2$-algebras of $\text{THH}(\mathbb{F}_p)$ with $\mathbb{F}_p[\Omega S^3]$, the group ring of the $E_1$-group $\Omega S^3$ over $\mathbb{F}_p$, we use trace methods to compute its algebraic $K$-theory. 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.