Triangulated Categories Admitting Linear Generators

Abstract

The main result of this paper is that there is an additive equivalence between $\overline{\mathcal{C}}_n$, the Paquette-Yildirim completion of the discrete cluster categories of Dynkin type $A_{\infty}$, and the perfect derived category of a certain DG algebra. This additive equivalence preserves some of the triangulated structure: it commutes with the suspension functor and preserves triangles with at least two indecomposable terms. In the process, we introduce the notion of a linear generator $G$ in a Krull-Schmidt, Hom-finite triangulated category. It turns out that the existence of a linear generator affords a large amount of control over $\mathcal{T}$. For example, it allows us to describe all indecomposable objects in $\mathcal{T}$ in terms of $G$, to determine all triangles of $\mathcal{T}$ with at least two indecomposable objects, and to show that the Rouquier dimension of $\mathcal{T}$ is at most one. Moreover, we prove that there is an additive equivalence (which preserves some of the triangulated structure) between $\mathcal{T}$ and the perfect derived category of a certain DG algebra. Finally, we show that any triangulated category with a linear generator is additively equivalent to a thick subcategory of $\overline{\mathcal{C}}_n$.