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 ${\mathcal{D}}_{A\mathrm{/}k}$ is an isomorphism. This result generalizes the characteristic 0 case in which the ring ${\mathcal{D}}_{A\mathrm{/}k}$ appears as the enveloping algebra of the Lie-Rinehart algebra of the usual $k$-derivations of $A$ provided that $A$ is smooth over $k$.