  In this paper, we show that the homotopy category of N-complexes of
projective R-modules is triangle equivalent to the homotopy category of
projective T_{N-1}(R)- modules where T_{N-1}(R) is the ring of triangular
matrices of order N-1 with entries in R. We also define the notions of
N-singularity category and N-totally acyclic complexes. We show that the
category of N-totally acyclic complexes of finitely generated projective
R-modules embeds in the N-singularity category, which is a result analogous to
the case of ordinary chain complexes.
