On the Enlargement by Pr\"ufer Objects of the Cluster Category of type $A_\infty$

Abstract

In a paper by Holm and Jorgensen, the cluster category $\mathscr{D}$ of type $A_\infty$, with Auslander-Reiten quiver $\mathbb{Z} A_\infty$, is introduced. Slices in the Auslander-Reiten quiver of $\mathscr{D}$ give rise to direct systems; the homotopy colimit of such direct systems can be computed and these "Pr\"ufer objects" can be adjoined to form a larger category. It is this larger category, $\overline{\mathscr{D}},$ which is the main object of study in this paper. We show that $\overline{\mathscr{D}}$ inherits a nice geometrical structure from $\mathscr{D}$; "arcs" between non-neighbouring integers on the number line correspond to indecomposable objects, and in the case of $\overline{\mathscr{D}}$ we also have arcs to infinity which correspond to the Pr\"ufer objects. During the course of this paper, we show that $\overline{\mathscr{D}}$ is triangulated, compute homs, investigate the geometric model, and we conclude by computing the cluster tilting subcategories of $\overline{\mathscr{D}}$.