Steenrod operations on the de Rham cohomology of algebraic stacks

TL;DR

This paper extends the theory of Steenrod operations to the de Rham cohomology of algebraic stacks in characteristic p, providing explicit computations for classifying stacks of various groups.

Contribution

It introduces and studies mod p Steenrod operations on de Rham cohomology of algebraic stacks, including explicit calculations for specific group classifying stacks.

Findings

Defined Steenrod operations on de Rham cohomology of algebraic stacks.

Computed the action of these operations on classifying stacks of finite, reductive, and orthogonal groups.

Extended previous work by Epstein, May, and Drury to a new algebraic geometric setting.

Abstract

Building up on work of Epstein, May and Drury, we define and investigate the mod $p$ Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$. We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which $p$ is not a torsion prime, and (special) orthogonal groups when $p=2$.