Torsion pairs were introduced by Dickson in 1966 as a generalization of the concept of torsion abelian group to arbitrary abelian categories. Using torsion pairs, we can divide complex abelian categories in smaller parts which are easier to understand. In this thesis we discuss torsion pairs in the category of modules over a finite-dimensional algebra, in particular we explore the relation between torsion pairs in the category of all modules and torsion pairs in the category of finite-dimensional modules. In the second chapter of the thesis, we present the analogue of a classical theorem of Auslander in the context of τ-tilting theory: for a finite-dimensional algebra the number of torsion pairs in the category of finite-dimensional modules is finite if and only if every brick over such algebra is finite- dimensional. In the third chapter, we revisit the Ingalls-Thomas correspondences between torsion pairs and wide subcategories in the context of large torsion pairs. We provide a nice description of the resulting wide subcategories and show that all such subcategories are coreflective. In the final chapter, we describe mutation of cosilting modules in terms of an operation on the Ziegler spectrum of the algebra.

On large and small torsion pairs / Sentieri, Francesco. - (2022 Jun 30), pp. 1-101. [10.15168/11572_348239]

### On large and small torsion pairs

#### Abstract

Torsion pairs were introduced by Dickson in 1966 as a generalization of the concept of torsion abelian group to arbitrary abelian categories. Using torsion pairs, we can divide complex abelian categories in smaller parts which are easier to understand. In this thesis we discuss torsion pairs in the category of modules over a finite-dimensional algebra, in particular we explore the relation between torsion pairs in the category of all modules and torsion pairs in the category of finite-dimensional modules. In the second chapter of the thesis, we present the analogue of a classical theorem of Auslander in the context of τ-tilting theory: for a finite-dimensional algebra the number of torsion pairs in the category of finite-dimensional modules is finite if and only if every brick over such algebra is finite- dimensional. In the third chapter, we revisit the Ingalls-Thomas correspondences between torsion pairs and wide subcategories in the context of large torsion pairs. We provide a nice description of the resulting wide subcategories and show that all such subcategories are coreflective. In the final chapter, we describe mutation of cosilting modules in terms of an operation on the Ziegler spectrum of the algebra.
##### Scheda breve Scheda completa Scheda completa (DC)
30-giu-2022
XXXIV
2019-2020
Matematica (29/10/12-)
Mathematics
Anegeleri Hugel, Lidia
no
Inglese
Settore MAT/02 - Algebra
File in questo prodotto:
File
On large and small torsion pairs.pdf

accesso aperto

Tipologia: Tesi di dottorato (Doctoral Thesis)
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11572/348239`
• ND
• ND
• ND