nLab graded derivation

Idea

The term graded derivation would a priori refer in general to suitably compatible derivations of graded algebras, but is in practice used mostly for those operations $d \,\colon\, A \to A$ on superalgebras $A$ that satisfy the condition that for $a_1, a_2 \,\in\, A$ a pair of elements each of homogeneous degree $deg(a_i) \in \mathbb{Z} \,mod\, 2$ the derivation of their product is

$$d(a_1 \cdot a_2) \;=\; (d a_1) \cdot a_2 + (-1)^{deg(a_1) \cdot deg(d)} a_1 \cdot d a_2 \,,$$

where $deg(d) \,\in\, \mathbb{Z} \,mod\,2$ is the degree of the graded derivation.

For $d = 0$ this is the law satisfied by an ordinary (un-graded) derivation, also known as the Leibniz rule satisfied by ordinary differentiation. The archetypical example of a non-trivially graded derivation is the de Rham differential acting on the de Rham algebra.