THE HOMOTOPY CATEGORIES OF N-COMPLEXES OF INJECTIVES AND PROJECTIVES

We investigate the homotopy category ${\mathcal{K}}_N(Inj{\mathfrak{A}})$ of N-complexes of injectives in a Grothendieck abelian category ${\mathfrak{A}}$ not necessarily locally noetherian, and prove that the inclusion ${\mathcal{K}}_N(Inj{\mathfrak{A}}){\rightarrow}{\mathcal{K}}({\mathfrak{A}})$ has a left adjoint and ${\mathcal{K}}_N(Inj{\mathfrak{A}})$ is well generated. We also show that the homotopy category ${\mathcal{K}}_N(PrjR)$ of N-complexes of projectives is compactly generated whenever R is right coherent.