Towards the dual motivic Steenrod algebra in positive characteristic

Abstract

The dual motivic Steenrod algebra with mod $\ell$ coefficients was computed by Voevodsky over a base field of characteristic zero, and by Hoyois, Kelly, and {\O}stv{\ae}r over a base field of characteristic $p \neq \ell$. In the case $p = \ell$, we show that the conjectured answer is a retract of the actual answer. We also describe the slices of the algebraic cobordism spectrum $MGL$: we show that the conjectured form of $s_n MGL$ is a retract of the actual answer.