The existence of homotopy resolutions of $N$-complexes

Xiaoyan Yang ; Junpeng Wang

Homology, homotopy, and applications, Tome 17 (2015) no. 2, pp. 291-316.

Résumé

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 $\mathcal{D}_N(R)$.

Détail

DOI : 10.4310/HHA.2015.v17.n2.a14

Classification : 18E30, 18G10, 18G35
Keywords: $N$-complex, homotopy category, homotopy resolution, recollement