Journal of the
Korean Mathematical Society JKMS

We investigate the homotopy category $\mathcal{K}_N(\mathrm{Inj}\mathscr{A})$ of $N$-comp\-lexes of injectives in a Grothendieck abelian category $\mathscr{A}$ not necessarily locally noetherian, and prove that the inclusion $\mathcal{K}(\mathrm{Inj}\mathscr{A})\rightarrow\mathcal{K}(\mathscr{A})$ 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: 16E05, 16E10

Supported by: This research was partially supported by National Natural Science Foundation of China (11761060) and Improvement of Young
Teachers Scientific Research Ability (NWNU-LKQN-16-5).