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

TL;DR

This paper investigates the algebraic $K$-theory of the topological Hochschild homology of finite fields, revealing its graded structure, computation methods, and uniqueness as an $E_2$-ring, with broader implications for perfect fields of characteristic $p$.

Contribution

It identifies the graded structure of $ ext{THH}( ext{F}_p)$, computes its algebraic $K$-theory using trace methods, and shows its uniqueness as an $E_2$-ring determined by homotopy groups.

Findings

Computed algebraic $K$-theory of $ ext{THH}( ext{F}_p)$

Established the graded structure from cyclic bar construction

Proved $ ext{THH}( ext{F}_p)$ is uniquely determined by homotopy groups

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.