An exact structure approach to almost rigid modules over quivers of type $\mathbb{A}$

TL;DR

This paper introduces a new exact structure on the module category of type A quivers, linking maximal almost rigid modules to tilting modules and polygon triangulations, with implications for generalized quiver types.

Contribution

It presents a novel exact structure that equates maximal almost rigid modules with tilting modules, providing a new perspective on their classification and properties.

Findings

Maximal almost rigid modules correspond to tilting modules under the new exact structure.

The module category becomes a 0-Auslander category with the new structure.

Generalizations to type D and gentle algebras are discussed.

Abstract

Let $A$ be the path algebra of a quiver of Dynkin type $\mathbb{A}_n$. The module category $\text{mod}\,A$ has a combinatorial model as the category of diagonals in a polygon $S$ with $n+1$ vertices. The recently introduced notion of almost rigid modules is a weakening of the classical notion of rigid modules. The importance of this new notion stems from the fact that maximal almost rigid $A$-modules are in bijection with the triangulations of the polygon $S.$ In this article, we give a different realization of maximal almost rigid modules. We introduce a non-standard exact structure $\mathcal{E}_\diamond$ on $\text{mod}\,A$ such that the maximal almost rigid $A$-modules in the usual exact structure are exactly the maximal rigid $A$-modules in the new exact structure. A maximal rigid module in this setting is the same as a tilting module. Thus the tilting theory relative to the exact structure $\mathcal{E}_\diamond$ translates into a theory of maximal almost rigid modules in the usual exact structure. As an application, we show that with the exact structure $\mathcal{E}_\diamond$, the module category becomes a 0-Auslander category in the sense of Gorsky, Nakaoka and Palu. We also discuss generalizations to quivers of type $\mathbb{D}$ and gentle algebras.