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# Homology, Homotopy and Applications

## Volume 17 (2015)

### Number 2

### The existence of homotopy resolutions of $N$-complexes

Pages: 291 – 316

DOI: http://dx.doi.org/10.4310/HHA.2015.v17.n2.a14

#### Authors

#### Abstract

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 $dg$-projective resolutions for arbitrary $N$-complexes. As applications of these results, we prove that the category $\mathcal{D}_N(R)$ is compactly generated, the category $\mathcal{K}_N(\mathscr{I})$ of injectives is compactly generated whenever $R$ is left noetherian, and the category $\mathcal{K}_N(\mathscr{P})$ 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 $\mathcal{K}_N(R)$ relative to $\mathcal{K}^{ex}_N(R)$ and $\mathcal{D}_N(R)$.

#### Keywords

$N$-complex, homotopy category, homotopy resolution, recollement

#### 2010 Mathematics Subject Classification

18E30, 18G10, 18G35