The topological Hochschild homology of algebraic $K$-theory of finite fields

Abstract

Let $K(\mathbb{F}_q)$ be the algebraic $K$-theory spectrum of the finite field with $q$ elements and let $p \geq 5$ be a prime number coprime to $q$. In this paper we study the mod $p$ and $v_1$ topological Hochschild homology of $K(\mathbb{F}_q)$, denoted $V(1)_*THH(K(\mathbb{F}_q))$, as an $\mathbb{F}_p$-algebra. The computations are organized in four different cases, depending on the mod $p$ behaviour of the function $q^n-1$. We use different spectral sequences, in particular the B\"okstedt spectral sequence and a generalization of a spectral sequence of Brun developed in an earlier paper. We calculate the $\mathbb{F}_p$-algebras $THH_*(K(\mathbb{F}_q); H\mathbb{F}_p)$, and we compute $V(1)_*THH(K(\mathbb{F}_q))$ in the first two cases.