graded derivation (Rev #2, changes)

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If we have a graded algebra $A$, and $D$ is a homogeneous linear map of grade $d=|\mathrm{deg}(D|)$ d =~~ |D|~~ deg(D) on $A$ then $D$ is a homogeneous derivation if

$$D(\mathrm{ab})=D(a)b+{\u03f5}^{|\mathrm{deg}(a|)|\cdot \mathrm{deg}(D|)}\mathrm{aD}(b)$$~~ D(ab)=D(a)b+\epsilon^{|a||D|}aD(b)~~ D(ab)=D(a)b+\epsilon^{deg(a)\cdot deg(D)}aD(b)

$\epsilon\in\{-1,1\}$ acting on homogeneous elements of $A$ . A graded derivation is a sum of homogeneous derivations with the same$\epsilon$.

If~~ the~~~~ commutator~~~~ factor~~$\epsilon = 1$ , this definition reduces to the~~ usual~~ Leibniz~~ case.~~ rule.~~ If~~~~$\epsilon = -1$~~~~, however, then ~~~~$D(ab)=D(a)b+(-1)^{|a|}aD(b)$~~~~, for odd ~~~~$|D|$~~~~. They are called anti-derivations.~~

If $\epsilon = -1$, then $D(ab)=D(a)b+(-1)^{|a|}aD(b)$, for odd $|D|$. The notion of graded derivations of odd degree is sometimes called *antiderivation* or *anti-derivation* or *integration*.

Examples of anti-derivations include the exterior derivative? and the interior product? acting on differential form?s.