  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 $\sum q^i \dim(C^i)$.
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.
