Well Generatedness and Adjoints for Homotopy Categories of N-complexes

Abstract

Let R be a ring and N ge 2. First, we prove that any deconstructible class of modules {mathcal {F}} over R induces two coreflective subcategories of the homotopy category textbf{K}_N(mathrm {Mod-}{R}) of (unbounded) N-complexes of right R-modules: the one whose objects are all N-complexes with components in {mathcal {F}}, textbf{K}_N({mathcal {F}}), and the one whose objects are the N-acyclic complexes with components in {mathcal {F}}, textbf{A}_N({mathcal {F}}). Second, we prove that for any decomposable class of modules mathcal G, the homotopy category of N-complexes, textbf{K}_N(mathcal G), is well generated. In particular, the homotopy category of N-complexes of projective modules is aleph _1-generated, which extends the well known result of Neeman for N=2.