Towards the dual motivic Steenrod algebra in positive characteristic

TL;DR

This paper investigates the structure of the dual motivic Steenrod algebra in positive characteristic, showing that certain conjectured forms are retracts of the actual algebra, extending known results from characteristic zero.

Contribution

It demonstrates that in positive characteristic, the conjectured dual motivic Steenrod algebra and slices of the algebraic cobordism spectrum are retracts of their actual counterparts.

Findings

The conjectured dual motivic Steenrod algebra is a retract of the actual algebra in characteristic p.

Slices of the algebraic cobordism spectrum $MGL$ are retracts of the actual slices.

Results extend the understanding of motivic Steenrod algebra beyond characteristic zero.

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.