Torsors and the Quillen-Barr-Beck cohomology for restricted Lie algebras

TL;DR

This paper develops an eight-term exact sequence for Quillen-Barr-Beck cohomology of restricted Lie algebras using torsors theory, with applications to algebraic group extensions over fields of prime characteristic.

Contribution

It introduces a new eight-term exact sequence connecting torsors and cohomology for restricted Lie algebras, extending previous theoretical frameworks.

Findings

Established an eight-term exact sequence for Quillen-Barr-Beck cohomology.

Connected algebraic group extensions to restricted Lie algebra cohomology.

Applied torsors theory to derive new cohomological relationships.

Abstract

Given an exact sequence of restricted Lie algebras using Duskin's torsors theory, we establish an eight term exact sequence for Quillen-Barr-Beck cohomology of restricted Lie algebras. As an application we obtain for any extension of algebraic groups over an algebraic closed field of prime characteristic an eight term exact sequence for the corresponding restricted Lie algebras extension.