Higher derivations of modules and the Hasse-Schmidt module

TL;DR

This paper explores higher derivations of modules, connecting classical concepts with recent advances in arc space differentials, and demonstrates that Hasse-Schmidt algebra functors commute, leading to new insights.

Contribution

It links Ribenboim's higher derivations with De Fernex and Docampo's work, deriving formulas for Kähler differentials via functor commutativity.

Findings

Derived the formula for Kähler differentials of the Hasse-Schmidt algebra

Established the commutativity of Hasse-Schmidt algebra functors

Connected classical higher derivations with modern arc space theory

Abstract

In this paper we revisit Ribenboim's notion of higher derivations of modules and relate it to the recent work of De Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the K\"ahler differentials of the Hasse-Schmidt algebra as a consequence of the fact that the Hasse-Schmidt algebra functors commute.