Some classes of subcategories of module categories: classifications and the relation between them
Date:
Conference Talk in The 53rd Symposium on Ring Theory and Representation Theory at Online (Yamaguchi University)
Based on my papers
- [1] Monobrick, a uniform approach to torsion-free classes and wide subcategories
- [2] Rigid modules and ICE-closed subcategories in quiver representations
- [3] ICE-closed subcategories and wide τ-tilting modules (joint with A. Sakai)
and my paper in preparation
- Computing various objects of an algebra from the poset of torsion classes
Abstract
In the representation theory of algebras, the study of subcategories of module categories have been one of the main topics. Among them, torsion classes (subcategories closed under quotients and extensions) have been attracted an attention. There are another class of subcategories of module categories: wide subcategories (subcategories closed under kernels, cokernels, and extensions). They are related to ring epimorphisms of an algebra, and the relation between torsion classes and wide subcategories have been studied by several authors.
Recently, I introduced some classes of subcategories of module categories which generalizes both torsion classes (or torsion-free classes) and wide subcategories: ICE-closed subcategories (Image-Cokernel-Extension-closed), and its dual IKE-closed subcategories (Image-Kernel-Extension-closed), hearts of intervals of torsion classes. They are particular classes of extension-closed subcategories which can be controlled by torsion classes. In this talk, I will talk about the classification results of these subcategories, and discuss the relation between these classes of subcategories, based on [1, 2, 3]. In particular, we can recover the poset structure of these subcategories using only the poset structure of torsion classes.