Abstract

Let $K\left({\mathbb{𝔽}}_q\right)$ be the algebraic $K$-theory spectrum of the finite field with $q$ elements and let $p\ge 5$ be a prime number coprime to $q$. We study the mod $p$ and $v_1$ topological Hochschild homology of $K\left({\mathbb{𝔽}}_q\right)$, denoted $V{\left(1\right)}_{\ast }THH\left(K\left({\mathbb{𝔽}}_q\right)\right)$, as an ${\mathbb{𝔽}}_p$-algebra. The computations are organized in four different cases, depending on the $p$-adic behavior of the function $q^n-1$. We use several 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 ${\mathbb{𝔽}}_p$-algebra ${THH}_{\ast }\left(K\left({\mathbb{𝔽}}_q\right);H{\mathbb{𝔽}}_p\right)$, and we compute $V{\left(1\right)}_{\ast }THH\left(K\left({\mathbb{𝔽}}_q\right)\right)$ in the first two cases.

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