= Graded derivation
{title2=$D(ab)=D(a)b+(-1)^{r|a|}aD(b)$}
A graded derivation of degree $r$ on a <graded algebra> $A$ is a linear map $D:A^n\to A^{n+r}$ satisfying
$$
D(ab)=D(a)b+(-1)^{r|a|}aD(b)
$$
for 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 $D^2(a)b+aD^2(b)$. 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.
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