A characterization of 2-fold torsion classes induced by $τ$-rigid modules

TL;DR

The work extends torsion theory by introducing $n$-fold torsion(-free) classes and shows that, over finite-dimensional algebras, $τ$-rigid modules yield $2$-fold torsion classes with a precise characterization. Central to the results is a bijection between basic $τ$-rigid modules and $2$-fold torsion classes satisfying a condition that ensures functorial finiteness and Ext-progenerators, connecting to the AIR framework. The paper also demonstrates that in the hereditary case this characterization recovers Enomoto’s ICE-closed correspondence, and provides explicit examples illustrating the richness of $2$-fold torsion classes beyond the cokernel construction. Collectively, these findings deepen the understanding of multi-fold torsion structures in representation theory and extend τ-tilting techniques to higher-fold contexts with concrete homological criteria.

Abstract

We introduce $n$-fold torsion(-free) classes of an abelian category. These are a generalization of ordinary torsion(-free) classes in the sense that $1$-fold torsion(-free) classes coincide with torsion(-free) classes. In the category of finitely generated modules over a finite dimensional algebra, we can naturally construct $n$-fold torsion classes from $τ$-rigid modules. We characterize the $2$-fold torsion classes induced by $τ$-rigid modules.

Theorems & Definitions (149)

• Theorem 1.1: kobayashi2024ke
• Definition 1.2: Definition \ref{['def nfold torf']}
• Proposition 1.3: Proposition \ref{['prop nfold vs Kn-1Eclosed']} and the dual of it
• Theorem 1.4: Theorem \ref{['thm resolving']}
• Theorem 1.5: Theorem \ref{['thm charac of kernel closure']}
• Theorem 1.6: Adachi_Iyama_Reiten_2014
• Proposition 1.7: Proposition \ref{['prop coknU is n+1fold tors']}
• Theorem 1.8: Theorem \ref{['thm charac of 2tors']}
• Proposition 1.9: Proposition \ref{['prop relationship of AIR and mine']}
• Theorem 1.10: Theorem \ref{['thm properties of 2tors induced by taurigid']}