Hasse--Schmidt derivations, divided powers and differential smoothness

TL;DR

This paper establishes canonical embeddings and morphisms connecting differential operators, divided powers, and Hasse-Schmidt derivations in the context of commutative algebra, revealing structural relationships and commutative diagrams.

Contribution

It introduces canonical maps linking differential operators, divided power structures, and Hasse-Schmidt derivations, clarifying their interrelations in algebraic geometry.

Findings

Canonical embedding of gr D into the graded dual of symmetric algebra of differentials.

Existence of a canonical morphism from divided power algebra of Hasse-Schmidt derivations to gr D.

Commutative diagram relating the constructed morphisms.

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.