Image-extension-closed subcategories of module categories of hereditary algebras

(joint work with A. Sakai)

Published in J. Pure Appl. Algebra, September 2023. Links: journal, arXiv:2208.13937

Citation: H. Enomoto, A. Sakai, Image-extension-closed subcategories of module categories of hereditary algebras, J. Pure Appl. Algebra 227 (2023), no. 9, Paper No. 107372.

Comment

We study subcategories of the module category of an artin algebra closed under taking images and extensions: IE-closed subcategories. They are precisely intersections of some torsion classes and some torsion-free classes. For the hereditary case, we classify IE-closed subcategories using Ext-projectives and Ext-injectives: a pair $(P,I)$ is twin rigid if there are exact sequences

\[0 \to P_0 \to P_1 \to I \to 0,\] \[0 \to P \to I^0 \to I^1 \to 0,\]

with $P_i \in \mathsf{add} P$ and $I^i \in \mathsf{add} I$. Then, for a hereditary artin algebra $\Lambda$, we show that IE-closed subcategories are in bijection with twin rigid $\Lambda$-modules by sending $\mathcal{C}$ to direct sum of all indecomposable Ext-projectives and Ext-injectives in $\mathcal{C}$, and sending $(P,I)$ to $\mathsf{Fac} P \cap \mathsf{Sub} I$.

For example:

  • $(P, D\Lambda)$ is twin rigid iff $P$ is tilting, and the corresponding IE-closed subcat is $\mathsf{Fac} P$.
  • $(\Lambda, I)$ is twin rigid iff $I$ is cotilting, and the corresponding IE-closed subcat is $\mathsf{Sub} I$.
  • $(P,P)$ is twin rigid iff $P$ is rigid, and the corresponding IE-closed subcat is $\mathsf{add} P$.

We also develop mutation of twin rigid modules, which enables us to compute all twin rigid modules if $\Lambda$ is representation-finite.

Presentation materials