The homotopy categories of $N$-complexes of injectives and projectives
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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.
Abstract
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.
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