A geometric model for the non-$\tau$-rigid modules of type $\widetilde{D}_n$

TL;DR

This paper introduces a geometric model for non-$ au$-rigid modules over acyclic path algebras of type $ ilde{D}_n$, enabling combinatorial computation of extension spaces via intersection formulas.

Contribution

It extends geometric models to non-$ au$-rigid modules of type $ ilde{D}_n$, incorporating decorations to handle infinitely many stable tubes.

Findings

Provides intersection-dimension formulas for $ ilde{D}_n$ modules.

Enables combinatorial calculation of homological data.

Introduces decorated geometric models to distinguish modules in different stable tubes.

Abstract

We give a geometric model for the non-$\tau$-rigid modules over acyclic path algebras of type $\widetilde{D}_n$. Similar models have been provided for module categories over path algebras of types $A_n, D_n,$ and $\widetilde{A}_n$ as well as the $\tau$-rigid modules of type $\widetilde{D}_n$. A major draw of these geometric models is the "intersection-dimension formulas" they often come with. These formulas give an equality between the intersection number of the curves representing the modules in the geometric model and the dimension of the extension spaces between the two modules. This formula allows us to calculate the homological data between two modules combinatorially. Since there are infinitely many distinct homogeneous stable tubes in the regular component of the Auslander-Reiten quiver of type $\widetilde{D}_n$, all of which are disjoint, our geometric data requires an extra decoration on the admissible edges in our geometric model to prevent intersections between curves corresponding to modules in distinct stable tubes of the Auslander-Reiten quiver.