Hasse-Schmidt derivations, divided powers and differential smoothness

Let k be a commutative ring, A a commutative k-algebra and D the filtered ring of k-linear differential operators of A. We prove that: (1) The graded ring gr D admits a canonical embedding θ into the graded dual of the symmetric algebra of the module ΩA/k of differentials of A over k, which has a canonical divided power structure. (2) There is a canonical morphism ϑ from the divided power algebra of the module of k-linear Hasse-Schmidt integrable derivations of A to gr D. (3) Morphisms θ and ϑ fit into a canonical commutative diagram.Ministerio de Educación y CienciaFondo Europeo de Desarrollo Regiona