Graded derivation ($D(ab)=D(a)b+(-1{)}^{r|a|}aD(b)$)

A graded derivation of degree $r$ on a graded algebra $A$ is a linear map $D:A^n\to A^{n+r}$ satisfyingfor homogeneous $a,b$. Degree zero gives the ordinary Leibniz rule. For odd $r$, the sign is $(-1{)}^{|a|}$, as for the differential in a differential graded algebra.

If $D$ is odd, its square is an even derivation: expanding $D^2(ab)$ cancels the two mixed terms and leaves $D^2(a)b+a{D}^2(b)$. Consequently, proving $D^2=0$ on algebra generators proves it everywhere. This is the algebraic reason that nilpotence of an odd differential can be checked on generators; a general graded derivation need not itself square to zero.