Abstract
This thesis explores aspects of homological algebra within subcategories. It comprises of three papers, each in its own chapter.
In Chapter I, we investigate the homological algebra of proper abelian subcategories within a triangulated category equipped with a Serre functor. By approximating the Serre functor, we construct Nakayama functors, which in turn enable the definition of the Auslander-Reiten translations. We show that suitable proper abelian subcategories are dualising k-varieties and have enough projectives if and only if they have enough injectives. This framework yields a new proof for the existence of Auslander-Reiten sequences in the category of finite dimensional modules over a finite dimensional algebra.
Chapter II introduces a theory of rank functions on (d+2)-angulated categories, generalising the notion of rank function on triangulated categories introduced by Chuang and Lazarev. We establish a bijective correspondence between object-defined and morphismdefined rank functions. Inspired by work of Conde, Gorsky, Marks and Zvonareva, we
further demonstrate a bijective correspondence between rank functions on an Amiot-Lin (d+2)-angulated categories and certain additive functions on its associated module category. This leads to a decomposition theorem: integral rank functions admit a factorisation into irreducible components in this setting.
Chapter III studies the homological theory of differential modules via the Q-shaped derived category introduced by Holm and Jørgensen. We prove a differential module analogue of a classical result characterising when a finitely generated module over a local commutative noetherian ring has finite injective dimension. As an application, we provide a new characterisation of local Cohen-Macaulay rings using differential modules, offering an alternative perspective on a question originally posed by Bass.
In Chapter I, we investigate the homological algebra of proper abelian subcategories within a triangulated category equipped with a Serre functor. By approximating the Serre functor, we construct Nakayama functors, which in turn enable the definition of the Auslander-Reiten translations. We show that suitable proper abelian subcategories are dualising k-varieties and have enough projectives if and only if they have enough injectives. This framework yields a new proof for the existence of Auslander-Reiten sequences in the category of finite dimensional modules over a finite dimensional algebra.
Chapter II introduces a theory of rank functions on (d+2)-angulated categories, generalising the notion of rank function on triangulated categories introduced by Chuang and Lazarev. We establish a bijective correspondence between object-defined and morphismdefined rank functions. Inspired by work of Conde, Gorsky, Marks and Zvonareva, we
further demonstrate a bijective correspondence between rank functions on an Amiot-Lin (d+2)-angulated categories and certain additive functions on its associated module category. This leads to a decomposition theorem: integral rank functions admit a factorisation into irreducible components in this setting.
Chapter III studies the homological theory of differential modules via the Q-shaped derived category introduced by Holm and Jørgensen. We prove a differential module analogue of a classical result characterising when a finitely generated module over a local commutative noetherian ring has finite injective dimension. As an application, we provide a new characterisation of local Cohen-Macaulay rings using differential modules, offering an alternative perspective on a question originally posed by Bass.
| Original language | English |
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| Publication status | In preparation - 2025 |