Bounding The Orlov Spectrum For A Completion Of Discrete Cluster Categories

We classify thick subcategories in a Paquette-Y\i ld\i r\i m completion $\overline{\mathcal{C}}$ of a discrete cluster category of Dynkin type $A_{\infty}$. To do this we introduce the notion of homologically connected objects, and the hc (=homologically connected) decomposition of an object into homologically connected objects in a $\mathrm{Hom}$-finite, Krull-Schmidt triangulated category. We show that any object in a $\overline{\mathcal{C}}$ has a hc decomposition, and that the hc decomposition determines the thick closure of an object. Moreover, we use this result to classify the classical generators of $\overline{\mathcal{C}}$ as homologically connected objects satisfying a maximality condition. Every homologically connected object has an invariant, known as the homological length, and we show that in $\overline{\mathcal{C}}$ this homological length is an upper bound for the generation time of a classical generator. This allows us to provide an upper bound for the Orlov spectrum of $\overline{\mathcal{C}}$, as well as giving the Rouquier dimension.