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 The homotopy categories of N-complexes of injectives and projectives J. Korean Math. Soc.Published online 2019 Apr 03 Zongyang Xie, and Xiaoyan Yang Northwest Normal University Abstract : We investigate the homotopy category $\mathcal{K}_N(\mathrm{Inj}\mathscr{A})$ of $N$-complexes of injectives in a Grothendieck abelian category $\mathscr{A}$ not necessarily locally noetherian, and prove that the inclusion $\mathcal{K}_N(\mathrm{Inj}\mathscr{A})\rightarrow\mathcal{K}_N(\mathscr{A})$ has a left adjoint and $\mathcal{K}_N(\mathrm{Inj}\mathscr{A})$ is well generated. We also show that the homotopy category $\mathcal{K}_N(\mathrm{Prj}R)$ of $N$-complexes of projectives is compactly generated whenever $R$ is right coherent. Keywords : $N$-complex; homotopy category MSC numbers : 18G35; 18E30; 18G10 Full-Text :