Classifying Subcategories of Triangulated Categories

Abstract

The topic of this thesis is classification of subcategories of triangulated categories. We first state and prove the Hopkins-Neeman theorem, which gives a bijection between thick subcategories of the derived category of perfect complexes over a commutative noetherian ring and specialization closed subsets of the prime ideal spectrum. Next, we present Benson, Iyengar and Krause's approach to classification problems, which involves using a central ring action on a compactly generated triangulated category to define local cohomology functors. If the stratification conditions are satisfied, the notion of triangulated support yields classification of both thick and localizing subcategories. Finally, we use the BIK-approach to investigate the case of a quantum polynomial ring A in two variables. We show that a nice commutative subring of A acts centrally on D(A). In order to figure out if this action satisfies the stratification conditions, we consider the representation theory of certain quotients of A. The situation turns out to be more complicated than in the commutative setting, and we conclude that the central ring action satisfies the local-global principle, but not the minimality condition.