Abstract
This is an attempt to generalize some basic facts of homological algebra to the case of "complexes" in which the differential satisfies the condition
d
N
=0
instead of the usual
d
2
=0
. Instead of familiar sign factors, the constructions related to such "N-complexes" involve powers of q where q is a primitive Nth root of 1. We show that the homology (in a natural sense) of an N-complex is an
(N−1)
-complex which is
(N−1)
-exact, and the role of the Euler characteristic is played by the trigonometric sum
∑
q
i
dim(
C
i
)
. By q-deforming the de Rham differential we develop a version of the theory of differential forms which is coordinate-dependent but covariant with respect to a natural Hopf algebra. In particular, there is a meaningful formalism of connections with the curvature being an N-form given by the N th power of the covariant derivative. For
N=3
the expression for the curvature is very similar to the Chern-Simons functional. This text was written in 1991.