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.