On the higher topological Hochschild homology of ${\mathbb{F}}_p$ and commutative ${\mathbb{F}}_p$-group algebras

Abstract

We extend Torleif Veen's calculation of higher topological Hochschild homology ${\mathsf{THH}}_{\ast }^{[n]}({\mathbb{F}}_p)$ from $n\le 2p$ to $n\le 2p+2$ for $p$ odd, and from $n=2$ to $n\le 3$ for $p=2$. We calculate higher Hochschild homology ${\mathsf{HH}}_{\ast }^{[n]}(k[x])$ over $k$ for any integral domain $k$, and ${\mathsf{HH}}_{\ast }^{[n]}({\mathbb{F}}_p[x]/x^{p^{\ell }})$ for all $n>0$. We use this and étale descent to calculate ${\mathsf{HH}}_{\ast }^{[n]}({\mathbb{F}}_p[G])$ for all $n>0$ for any cyclic group $G$, and therefore also for any finitely generated abelian group $G$. We show a splitting result for higher $\mathsf{THH}$ of commutative ${\mathbb{F}}_p$-group algebras and use this technique to calculate higher topological Hochschild homology of such group algebras for as large an $n$ as ${\mathsf{THH}}_{\ast }^{[n]}({\mathbb{F}}_p)$ is known for.

Submitted

References in corpus (1)

• Detecting Periodic Elements in Higher Topological Hochschild Homology