Torsion pairs in recollements of abelian categories

Xin Ma; Zhaoyong Huang

doi:10.1007/s11464-018-0712-1

Torsion Pairs and Quasi-abelian Categories

Algebras and Representation Theory ◽

10.1007/s10468-020-10004-y ◽

2020 ◽

Author(s):

Aran Tattar

Keyword(s):

Abelian Categories ◽

Torsion Pairs

AbstractWe 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 quasi-abelian categories. As an application, we generalise the recently defined algebraic Harder-Narasimhan filtrations to quasi-abelian categories.

Download Full-text

Torsion pairs and filtrations in abelian categories with tilting objects

Journal of Algebra and Its Applications ◽

10.1142/s0219498815501212 ◽

2015 ◽

Vol 14 (08) ◽

pp. 1550121

Author(s):

Jason Lo

Keyword(s):

Abelian Category ◽

Projective Dimension ◽

Explicit Description ◽

Homological Dimension ◽

Abelian Categories ◽

Dimension 2 ◽

Tilting Object ◽

Category Of Modules ◽

Torsion Pairs ◽

Tilting Objects

Given a noetherian abelian k-category [Formula: see text] of finite homological dimension, with a tilting object T of projective dimension 2, the abelian category [Formula: see text] and the abelian category of modules over End (T) op are related by a sequence of two tilts; we give an explicit description of the torsion pairs involved. We then use our techniques to obtain a simplified proof of a theorem of Jensen–Madsen–Su, that [Formula: see text] has a three-step filtration by extension-closed subcategories. Finally, we generalize Jensen–Madsen–Su's filtration to the case where T has any finite projective dimension.

Download Full-text

Configurations in abelian categories—I: Basic properties and moduli stacks

Advances in Mathematics ◽

10.1016/j.aim.2005.04.008 ◽

2006 ◽

Vol 203 (1) ◽

pp. 194-255 ◽

Cited By ~ 22

Author(s):

Dominic Joyce

Keyword(s):

Basic Properties ◽

Moduli Stacks ◽

Abelian Categories

Download Full-text

Extension of matrix pencil reduction to abelian categories

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500627 ◽

2018 ◽

Vol 17 (04) ◽

pp. 1850062

Author(s):

Olivier Verdier

Keyword(s):

Normal Form ◽

Vector Space ◽

Decomposition Theorem ◽

Abelian Category ◽

Jordan Canonical Form ◽

Main Technique ◽

Abelian Categories ◽

Category Of Modules ◽

Space Case ◽

And Control

Matrix pencils, or pairs of matrices, are used in a variety of applications. By the Kronecker decomposition theorem, they admit a normal form. This normal form consists of four parts, one part based on the Jordan canonical form, one part made of nilpotent matrices, and two other dual parts, which we call the observation and control part. The goal of this paper is to show that large portions of that decomposition are still valid for pairs of morphisms of modules or abelian groups, and more generally in any abelian category. In the vector space case, we recover the full Kronecker decomposition theorem. The main technique is that of reduction, which extends readily to the abelian category case. Reductions naturally arise in two flavors, which are dual to each other. There are a number of properties of those reductions which extend remarkably from the vector space case to abelian categories. First, both types of reduction commute. Second, at each step of the reduction, one can compute three sequences of invariant spaces (objects in the category), which generalize the Kronecker decomposition into nilpotent, observation and control blocks. These sequences indicate whether the system is reduced in one direction or the other. In the category of modules, there is also a relation between these sequences and the resolvent set of the pair of morphisms, which generalizes the regular pencil theorem. We also indicate how this allows to define invariant subspaces in the vector space case, and study the notion of strangeness as an example.

Download Full-text

Piecewise Hereditary Triangular Matrix Algebras

Algebra Colloquium ◽

10.1142/s1005386721000134 ◽

2021 ◽

Vol 28 (01) ◽

pp. 143-154

Author(s):

Yiyu Li ◽

Ming Lu

Keyword(s):

Positive Integer ◽

Triangular Matrix ◽

Derived Categories ◽

Matrix Algebras ◽

Tensor Algebras ◽

Finite Dimensional ◽

Abelian Categories ◽

Upper Triangular Matrix ◽

Orbit Categories ◽

Finite Dimensional Algebras

For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

Download Full-text