On a Conjecture I for unirational algebraic groups over an imperfect field

TL;DR

This paper generalizes Serre's Conjecture I for unirational algebraic groups over imperfect fields, proving triviality of the first Galois cohomology set under certain cohomological conditions.

Contribution

It extends existing conjectures to a broader class of algebraic groups over imperfect fields, providing new cohomological results.

Findings

First Galois cohomology set of unirational algebraic groups is trivial when cohomological dimension ≤ 1.

Proposes a generalization of Serre's Conjecture I for imperfect fields.

Utilizes recent structural advancements in algebraic groups over such fields.

Abstract

Using the recent advancements in the structure of algebraic groups over imperfect fields, we propose a generalization of Serre's Conjecture I and of results that revolve around it. In particular, we prove that the first Galois cohomology set of any unirational algebraic group is always trivial if the cohomological dimension of the field is less or equal to 1 in Kato's sense.