Rings of differential operators as enveloping algebras of Hasse–Schmidt derivations.

Let k be a commutative ring and A a commutative k -algebra. In this paper we introduce the notion of enveloping algebra of Hasse–Schmidt derivations of A over k and we prove that, under suitable smoothness hypotheses, the canonical map from the above enveloping algebra to the ring of differential operators D A / k is an isomorphism. This result generalizes the characteristic 0 case in which the ring D A / k appears as the enveloping algebra of the Lie-Rinehart algebra of the usual k -derivations of A provided that A is smooth over k. [ABSTRACT FROM AUTHOR]