Hasse--Schmidt derivations, divided powers and differential smoothness

Abstract

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 $\theta$ into the graded dual of the symmetric algebra of the module ${\Omega }_{A/k}$ of differentials of $A$ over $k$, which has a canonical divided power structure. (2) There is a canonical morphism $\vartheta$ from the divided power algebra of the module of $k$-linear Hasse-Schmidt integrable derivations of $A$ to $\gr D$. (3) Morphisms $\theta$ and $\vartheta$ fit into a canonical commutative diagram.

Cited by in corpus (6)

• On the modules of m-integrable derivations in non-zero characteristic
• On Hasse--Schmidt derivations: the action of substitution maps
• Leaps of modules of integrable derivations in the sense of Hasse-Schmidt
• Rings of differential operators as enveloping algebras of Hasse--Schmidt derivations
• Hasse--Schmidt modules versus integrable connections
• On the bracket of integrable derivations