Rigid modules and ICE-closed subcategories in quiver representations

We introduce image-cokernel-extension-closed (ICE-closed) subcategories of module categories. This class unifies both torsion classes and wide subcategories. We show that ICE-closed subcategories over the path algebra of Dynkin type are in bijection with basic rigid modules and that ICE-closed subca...

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Bibliographic Details
Published inJournal of algebra Vol. 594; pp. 364 - 388
Main Author Enomoto, Haruhisa
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.03.2022
Subjects
Exceptional sequences
ICE-closed subcategories
Path algebra
Rigid modules
Rigid modules
Exceptional sequences
16G20
Path algebra
16G10
ICE-closed subcategories
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Summary:We introduce image-cokernel-extension-closed (ICE-closed) subcategories of module categories. This class unifies both torsion classes and wide subcategories. We show that ICE-closed subcategories over the path algebra of Dynkin type are in bijection with basic rigid modules and that ICE-closed subcategories are precisely torsion classes in some wide subcategories. We also study natural maps from ICE-closed subcategories to torsion classes and wide subcategories in terms of rigid modules. Finally, we prove that the number of ICE-closed subcategories does not depend on the orientation of the quiver, and give an explicit formula for each Dynkin type, which is equal to the large Schröder number for type A.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2021.12.008