On universal stably free modules in positive characteristic

TL;DR

This paper investigates the mod p motivic cohomology of specific homogeneous varieties in positive characteristic, analyzing Steenrod operations and demonstrating the non-existence of sections for certain quotient maps.

Contribution

It provides new results on the structure of motivic cohomology and the behavior of quotient maps in positive characteristic without restrictions on the base field.

Findings

Certain quotient maps do not admit sections in the studied cohomology.

Steenrod operations act non-trivially on the motivic cohomology of these varieties.

Results hold without restrictions on the characteristic of the base field.

Abstract

We study the mod $p$ motivic cohomology of homogeneous varieties such as $GL_{n}/GL_{r}$ or $Sp_{2n}/Sp_{2n-2}$ along with the action of the Steenrod operations, without restrictions on the characteristic of the base field. In particular, we prove that certain quotient maps do not admit sections.