Cohomology of Restricted Lie-Rinehart Algebras and the Brauer Group

TL;DR

This paper explores the cohomology of restricted Lie-Rinehart algebras and connects it to the Brauer group of purely inseparable extensions, providing new classification results for extensions.

Contribution

It introduces the category of restricted Lie-Rinehart algebras, defines their cotriple cohomology, and classifies extensions, linking cohomology to the Brauer group.

Findings

Defined cotriple cohomology groups for restricted Lie-Rinehart algebras

Classified restricted Lie-Rinehart extensions

Connected cohomology with the Brauer group of inseparable extensions

Abstract

We give an interpretation of the Brauer group of a purely inseparable extension of exponent 1, in terms of restricted Lie-Rinehart cohomology. In particular, we define and study the category $p$-$\rm{LR}(A)$ of restricted Lie-Rinehart algebras over a commutative algebra $A$. We define cotriple cohomology groups $H_{p-LR}(L,M)$ for $L\in p$-$\rm{LR}(A)$ and $M$ a Beck $L$-module. We classify restricted Lie-Rinehart extensions. Thus, we obtain a classification theorem for regular extensions considered by Hoshschild.