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 $p\ge 5$ 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 $V(1{)}_{\ast }THH(K({\mathbb{F}}_q))$, 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 Bökstedt spectral sequence and a generalization of a spectral sequence of Brun developed in an earlier paper. We calculate the ${\mathbb{F}}_p$-algebras $TH{H}_{\ast }(K({\mathbb{F}}_q);H{\mathbb{F}}_p)$, and we compute $V(1{)}_{\ast }THH(K({\mathbb{F}}_q))$ in the first two cases.

References in corpus (2)

• On the Brun spectral sequence for topological Hochschild homology
• On topological Hochschild homology of the $K(1)$-local sphere

Cited by in corpus (1)

• On the Brun spectral sequence for topological Hochschild homology