Hasse--Schmidt derivations versus classical derivations

TL;DR

This paper explores the relationship between multivariate Hasse--Schmidt derivations and classical derivations over commutative algebras, showing how to recover the former from the latter in characteristic zero.

Contribution

It generalizes previous work by establishing a method to derive multivariate Hasse--Schmidt derivations from classical derivations in arbitrary characteristic zero settings.

Findings

Hasse--Schmidt derivations can be associated with classical derivations.

In characteristic zero, the original multivariate Hasse--Schmidt derivation is recoverable from classical derivations.

The constructions extend previous results to more general algebraic contexts.

Abstract

In this paper we survey the notion and basic results on multivariate Hasse--Schmidt derivations over arbitrary commutative algebras and we associate to such an object a family of classical derivations. We study the behavior of these derivations under the action of substitution maps and we prove that, in characteristic $0$, the original multivariate Hasse--Schmidt derivation can be recovered from the associated family of classical derivations. Our constructions generalize a previous one by M. Mirzavaziri in the case of a base field of characteristic $0$.