Modulo $\tau^{p-1}$ motivic Hochschild homology of modulo $p$ motivic cohomology

TL;DR

This paper computes a specific form of motivic Hochschild homology over an algebraically closed field using the motivic Greenlees spectral sequence, focusing on modulo p motivic cohomology and a particular torsion element.

Contribution

It introduces a novel computation of the Hochschild homology of modulo p motivic cohomology in the stable motivic homotopy category using spectral sequences.

Findings

Explicit computation of MHH(MZ/p)/τ^{p-1} over algebraically closed fields

Application of the motivic Greenlees spectral sequence in this context

Advancement in understanding motivic Hochschild homology structures

Abstract

We use the motivic Greenlees spectral sequence from arXiv:2408.00338 to compute Hochschild homology in the stable motivic homotopy category over an algebraically closed field. Our target is $MHH(M\mathbb{Z}/p)/\tau^{p-1}$, where $M\mathbb{Z}/p$ is modulo $p$ motivic cohomology, $p$ a prime number different from the characteristic of the base.