# Motivic Hochschild homology

KA2W01 - Algebraic K-theory, motivic cohomology and motivic homotopy theory

I will introduce a motivic variant of Hochschild homology valued in the stable $\mathbb{A}^1$homotopy category. Over an algebraically closed field of characteristic distinct from $p$, computations reveal that the motivic Hochschild homology of $\mathbb{F}_p$ has a rich pattern of $\tau$-torsion. This places significant constraints on potential— even expected—- motivic analogues of classical and $C_2$-equivariant theorems (e.g. James splitting and motivic Eilenberg-Mac Lane spectra as Thom spectra).  This is joint work with Bj{\o}rn Dundas, Mike Hill, and Paul Arne {\O}stv{\ae}r.

This talk is part of the Isaac Newton Institute Seminar Series series.