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 $3\leqslant s\leqslant 5$. 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 $\tau_m$-closures of injective modules are realized as admissible $(m+2)$-partial triangulations, where $\tau_m$ are higher Auslander-Reiten translations with $2\leqslant m \leqslant n$. Finally, we provide a complete classification of gentle algebras that are $\tau_n$-finite or $n$-complete introduced by Iyama [I11].