Maximal rigid modules over a gentle algebra and applications to higher Auslander-Reiten theory

We construct a bijective correspondence between the set of rigid modules over a gentle algebra and the set of admissible arc systems on the associated coordinated-marked surface. In particular, a maximal rigid module aligns with an equivalence class of admissible $5$-partial triangulations, which is an (admissible) set of simple arcs dissecting the surface into s-gons with 3 s 5. Furthermore, the rank of the maximal rigid module is equal to the rank of the algebra plus the number of internal $4$-gons and $5$-gons in the associated $5$-partial triangulation. Subsequently, these results facilitate an exploration of the higher Auslander-Reiten theory for gentle algebras with global dimension n. The τₘ-closures of injective modules are realized as admissible $(m+2)$-partial triangulations, where τₘ are higher Auslander-Reiten translations with 2 m n. Finally, we provide a complete classification of gentle algebras that are τₙ-finite or n-complete introduced by Iyama [I11].