From Grothendieck groups to generators: the discrete cluster categories of type A∞

Murphy, David (2023) From Grothendieck groups to generators: the discrete cluster categories of type A∞. PhD thesis, University of Glasgow.

Abstract

In this thesis we look at two closely related families of categories: the discrete cluster categories of Dynkin type A∞, and their completions in the sense of Paquette and Yıldırım.

We compute the triangulated Grothendieck group of the discrete cluster categories of Dynkin type A∞, as well as their Paquette-Yıldırım completions. Further, we provide a counterexample to a theorem by Palu and provide a corrected statement of the result.

We also introduce the concept of homologically connected objects, and show that any object in the Paquette-Yıldırım completion of a discrete cluster category of Dynkin type A∞ can be decomposed into homologically connected direct summands, and that the smallest thick subcategory containing an object is determined by its decomposition into homologically connected direct summands. This allows us to classify the classical generators of the Paquette-Yıldırım completions of the discrete cluster categories of Dynkin type A∞, and associate an integer to each classical generator that is an upper bound on their generation time. This allows us to compute an upper bound for the Orlov spectrum, and to compute the Rouquier dimension of the Paquette-Yıldırım completions.

Further, we compute the graded endomorphism ring of a chosen classical generator as a Z graded, upper triangular matrix ring with polynomial rings and Laurent polynomial rings as entries.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Bellamy, Professor Gwyn, Gratz, Professor Sira and Stevenson, Dr. Greg
• Date of Award: 2023
• Depositing User: Theses Team
• Unique ID: glathesis:2023-83807
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 07 Sep 2023 11:30
• Last Modified: 07 Sep 2023 13:14
• Thesis DOI: 10.5525/gla.thesis.83807
• URI: https://theses.gla.ac.uk/id/eprint/83807