While (co)chain complexes involve $d^2 = 0$, (co)chain $N$-complexes involve a differential $d$ satisfying $d^N = 0$, where $N$ is a positive integer bigger than $2$.
The $N=3$ case has been studied in
The modern attention to the subject started in
M. M. Kapranov, On the q-analog of homological algebra, q-alg/9611005
Michel Dubois-Violette, Lectures on differentials, generalized differentials and on some examples related to theoretical physics, math.QA/0005256
Michel Dubois-Violette, Tensor product of N-complexes and generalization of graded differential algebras, Bulg. J. Phys. 36 (2009) 227–236 pdf
Michel Dubois-Violette, Marc Henneaux, Tensor fields of mixed Young symmetry type and N-complexes, math.QA/0110088 doi
In the following article the $N$-homological versions of Tor and Ext functors are expressed in terms of classical Tor and Ext (of course, with shifted indices). As a consequence of this type of result, the interest in the $N$-homological algebra somewhat diminished.
Last revised on March 7, 2013 at 01:43:19. See the history of this page for a list of all contributions to it.