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?



What should I call a "differential" which cubes, rather than squares, to zero?
Many years ago, when I was a graduate student, I remember seeing a couple of papers on the homologies of operators satisfying $\partial^p=0$, generalizing the case $p=2$. I seem to remember that they were by somebody like Steenrod, and it might evan have been in the Annals, sometime in the 40s or 50s. 


Unfortunately, I'm at home now and not able to access MathSciNet to look it up. However, I do remember that there was something like a set of axioms, generalizing the Steenrod axioms, for the various '$p$-homologies' that could be associated to topological spaces using such operators.


I forget now why I was interested in them. When I'm back in my office (maybe tomorrow), I'll try to find it on MathSciNet.


