The motivic Steenrod algebra in positive characteristic

TL;DR

This paper extends Voevodsky's result on the motivic Steenrod algebra to positive characteristic schemes by replacing topological methods with étale cohomology and alterations, broadening the applicability of the theory.

Contribution

It proves that the algebra of bistable operations in mod l motivic cohomology is generated by Steenrod operations over schemes in positive characteristic, removing the characteristic zero restriction.

Findings

The motivic Steenrod algebra is generated by Steenrod operations in positive characteristic.

The proof adapts Voevodsky's approach using étale cohomology and Gabber's alterations.

The result applies to essentially smooth schemes over fields of positive characteristic.

Abstract

Let S be an essentially smooth scheme over a field and l a prime number invertible on S. We show that the algebra of bistable operations in the mod l 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.