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

arXiv preprint, August 2022. Link: arXiv:2208.13937

Citation: H. Enomoto, A. Sakai, Image-extension-closed subcategories of module categories of hereditary algebras, arXiv:2208.13937.

## 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.