Bounding The Orlov Spectrum For A Completion Of Discrete Cluster Categories

TL;DR

This paper classifies thick subcategories in a completion of a discrete cluster category of Dynkin type A_infinity, introduces homologically connected objects and their decompositions, and uses these to bound the Orlov spectrum and Rouquier dimension.

Contribution

It introduces the notion of homologically connected objects and their decompositions, providing a new method to classify generators and bound the Orlov spectrum in this setting.

Findings

Classified thick subcategories in the completion of a discrete cluster category.

Established that homological length bounds the generation time of generators.

Provided an upper bound for the Orlov spectrum and Rouquier dimension.

Abstract

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.