Cohomology of restricted Lie–Rinehart algebras and the Brauer group

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-LR(A) of restricted Lie–Rinehart algebras over a commutative algebra A. We define cotriple cohomology groups Hp-LR(L,M) for L∈p-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 Hochschild.