nLab N-complex

Idea

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 $2$.

References

The $N=3$ case has been studied in

• W. Mayer, A new homology theory, I, II, Annals of Math. 43 (1942), 370–380, 594–605.

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.

• Christian Kassel, Marc Wambst, Algèbre homologique des N-complexes et homologie de Hochschild aux racines de l’unité, Publ. RIMS, Kyoto Univ. 34 (1998), p. 91-114 pdf q-alg/9705001

category: algebra