Homotopy category of N-complexes of projective modules

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 TN−1(R)TN−1(R)-modules where TN−1(R)TN−1(R) is the ring of triangular matrices of order N−1N−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.