Torsion Pairs and Quasi-abelian Categories
- PDF / 495,144 Bytes
- 25 Pages / 439.642 x 666.49 pts Page_size
- 37 Downloads / 188 Views
Torsion Pairs and Quasi-abelian Categories Aran Tattar1 Received: 21 April 2020 / Accepted: 6 October 2020 / © The Author(s) 2020
Abstract We define torsion pairs for quasi-abelian categories and give several characterisations. We show that many of the torsion theoretic concepts translate from abelian categories to quasiabelian categories. As an application, we generalise the recently defined algebraic HarderNarasimhan filtrations to quasi-abelian categories. Keywords Quasi-abelian category · Torsion class · Torsion pair · Harder-Narasimhan filtration Mathematics Subject Classification (2010) 18E05 · 18E10 · 19E40
1 Introduction Torsion classes were introduced for abelian categories by Dickson [11] to generalise the notion of torsion and torsionfree groups. Since then they have been widely studied in various contexts including (τ -)tilting theory [1, 16], lattice theory [12] and, more recently, stability conditions [8, 32]. Quasi-abelian categories are a particular class of exact categories (in the sense of Quillen [22]) whose maximal exact structure [27, 31] coincides with the class of all short exact sequences in the category (see Definition 2.1). As the name suggests, they are a weaker structure than abelian categories. Quasi-abelian categories appear naturally in cluster theory [29] and in the context of Bridgeland’s stability conditions [7]. Of particular interest to us is their appearence in torsion theory as observed by Bondal & Van den Bergh [10] and Rump [26]: Each torsion(free) class in an abelian category is quasi-abelian and every quasi-abelian category, Q, appears as the torsionfree class of a ‘left associated’ abelian category LQ and as a torsion class of an abelian category RQ . In this paper we seek to exploit this relationship to define and study torsion classes in quasi-abelian categories by describing torsion classes of quasi-abelian categories in terms of the torsion(free) classes in the associated abelian category. We note that torsion pairs in Presented by: Henning Krause Aran Tattar
[email protected] 1
School of Mathematics, University of Leicester, Leicester, LE1 7RH, UK
A. Tattar
pre-abelian and semi-abelian categories, which are weaker structures still than quasi-abelian categories, have been studied in [19]. In this more general context, torsion pairs no longer have the well-known characterisations that they have in the abelian set up. In [5] torsion theory in non-abelian, so-called homological categories has also been considered. Based on a characterisation of torsion pairs in abelian categories [11], we define a torsion pair for a quasi-abelian category as follows. Definition (Definition 2.4) Let Q be a quasi-abelian category. A torsion pair in Q is an ordered pair (T , F ) of full subcategories of Q satisfying the following. (a) HomQ (T , F ) = 0. (b) For all M in Q there exists a short exact sequence 0
T
M
M
MF
0
with T M ∈ T and MF ∈ F . In this case we call T a torsion class and F a torsionfree class. We establish a correspondence between certain torsion pairs in an abeli
Data Loading...
