The motivic Steenrod algebra in positive characteristic

Abstract

Let $S$ be an essentially smooth scheme over a field and $\ell \ne \mathrm{char}S$ a prime number. We show that the algebra of bistable operations in the mod $\ell$ motivic cohomology of smooth $S$-schemes is generated by the motivic Steenrod operations. This was previously proved by Voevodsky for $S$ a field of characteristic zero. We follow Voevodsky's proof but remove its dependence on characteristic zero by using etale cohomology instead of topological realization and by replacing resolution of singularities with a theorem of Gabber on alterations.