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

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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

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.

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