A subcategory with remarkable homological properties via actions of tensor categories

Abstract

By equipping the category $\mathcal{M}$ of modules over a finite dimensional algebra with the action of a finite tensor category $\mathcal{C}$, we introduce a subcategory $\mathcal{PM}$ satisfying numerous remarkable properties: it is generating and cogenerating, functorially finite, closed under Auslander--Reiten translations (i.e. precluster tilting), and it computes the finitistic dimension. Through a functor category perspective, $\mathcal{PM}$ is identified with the $\mathcal{C}$-module functors that are projective, and we also prove a $\mathcal{C}$-equivariant version of the Auslander(--Solberg) correspondence. As examples, we show that $\mathcal{PM}$ contains only finitely many indecomposable modules for some tensor product algebras. For some family of 1-Iwanaga--Gorenstein algebras introduced by Geiss--Leclerc--Schröer, we show that $\mathcal{PM}$ is exactly the subcategory of $τ$-locally free modules.