Hochschild homology of mod-$p$ motivic cohomology over algebraically closed fields

TL;DR

This paper computes Hochschild homology in the motivic setting over algebraically closed fields, revealing torsion classes linked to the mod-$p$ Steenrod algebra and connecting to topological Hochschild homology of finite fields.

Contribution

It introduces calculations of motivic Hochschild homology with torsion classes and relates these to classical topological results, expanding understanding of motivic homology.

Findings

Identification of torsion classes from the mod-$p$ Steenrod algebra

Connection to B"okstedt's topological Hochschild homology of finite fields

Extension of Hochschild homology calculations to motivic contexts

Abstract

We perform Hochschild homology calculations in the algebro-geometric setting of motives. The motivic Hochschild homology coefficient ring contains torsion classes which arise from the mod-$p$ motivic Steenrod algebra and from generating functions on the natural numbers with finite non-empty support. Under the Betti realization, we recover B\"okstedt's calculation of the topological Hochschild homology of finite prime fields.