Homotopy category of N-complexes of projective modules

TL;DR

This paper establishes an equivalence between the homotopy category of N-complexes of projective modules and a category of modules over a triangular matrix ring, introducing new concepts of N-singularity and N-totally acyclic complexes.

Contribution

It extends the theory of homotopy categories to N-complexes and introduces N-singularity and N-totally acyclic complexes, generalizing classical results.

Findings

Homotopy category of N-complexes is equivalent to a category over a triangular matrix ring.

N-totally acyclic complexes embed into the N-singularity category.

New notions of N-singularity and N-totally acyclic complexes are defined and analyzed.

Abstract

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.