nLab Hasse-Schmidt derivation

Idea

The notion of Hasse-Schmidt derivation, or higher deriviation is an extension of the notion of derivation.

Definition

Let $R$ be a commutative rig and $S$ a $R$-module. A Hasse-Schmidt derivation is a family of linear maps $(D^{n}:S \rightarrow S)_{n \ge 0}$ such that:

• $D^{0} = Id$
• $D^{n}(ab) = \underset{0 \le k \le n}{\sum}D^{k}(a)D^{n-k}(b)$

Note that $D^{1}$ is then a derivation in the usual sense.

References

The notion is explicitely defined in:

• Morris Weifeld: Purely Inseparable Extensions and Higher Derivations, Transactions of the American Mathematical Society, 116 (1965) 435-449 pdf

where it is said that it is due to the paper:

• Helmut Hasse: Noch eine Begründung der Theorie der höheren Differrentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten. (Nach einer brieflichen Mitteilung von F. K. Schmidt in Jena.), Reine Angew. Math. 177 (1937) 215-223 [paper]

which defines the Hasse-Schmidt derivative.