The existence of homotopy resolutions of $N$-complexes

In this paper complexes with N -nilpotent differentials are considered.We proceed by generalizing a defining property of injective and projective resolutions to define dg-injective and dg-projective N -complexes, and construct dg-injective and dgprojective resolutions for arbitrary N -complexes.As applications of these results, we prove that the category D N (R) is compactly generated, the category K N (I ) of injectives is compactly generated whenever R is left noetherian, and the category K N (P) of projectives is compactly generated whenever R is a right coherent ring for which every flat left R-module has finite projective dimension.We also establish a recollement of the category K N (R) relative to K ex N (R) and D N (R).