Cohomology of restricted Lie–Rinehart algebras and the Brauer group

We give an interpretation of the Brauer group of a purely inseparable extension of exponent 11, in terms of restricted Lie–Rinehart cohomology. In particular, we define and study the category pp-LR(A) of restricted Lie–Rinehart algebras over a commutative algebra AA. We define cotriple cohomology groups Hp-LR(L,M)Hp-LR(L,M) for L∈pL∈p-LR(A) and MM a Beck LL-module. We classify restricted Lie–Rinehart extensions. Thus, we obtain a classification theorem for regular extensions considered by Hochschild.