If I had a vector space with a linear endomorphism $D$ satisfying $D^2 = 0$, I might call it a *differential* and study its *(co)homology* $\operatorname{ker}(D) / \operatorname{im}(D)$. I might say that $D$ is *exact* if this (co)homology vanishes. I would especially do this if $D$ increased by 1 some grading on my vector space.

But I don't have this structure. Instead, I have a vector space with an endomorphism $D$, which increases by 1 some grading, satisfying $D^3 = 0$ but $D^2 \neq 0$. Then there are two possible "(co)homologies": $\operatorname{ker}(D) / \operatorname{im}(D^2)$ and $ \operatorname{ker}(D^2)/\operatorname{im}(D) $. It is an amusing exercise that if one of these groups vanishes, then so does the other, so that it makes sense to talk about $D$ being "exact".

Surely this type of structure has appeared before. Does it have a standard name? Where can I read some elementary discussion?

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