On the Combinatorics of Gentle Algebras
Thomas Brüstle, Guillaume Douville, Kaveh Mousavand, Hugh Thomas, Emine Yıldırım
Abstract
For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for $\operatorname{Hom}(X,\tau Y)$ . We use this to describe support $\tau$ -tilting modules for $A$ . We give a combinatorial realization of maps in both directions realizing the bijection between support $\tau$ -tilting modules and functorially finite torsion classes. We give an explicit basis of $\operatorname{Ext}^{1}(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville [McC], showing that many but not all of them can be extended to general gentle algebras.
中文速览
针对温和代数（gentle algebra）上弦模（string module）之间的同态空间计算难题，这项工作通过将温和代数嵌入一个更大的"边缘代数"（fringed algebra），给出了 $\operatorname{Hom}(X, \tau Y)$ 的组合基底，从而绕开了 Auslander-Reiten 平移 $\tau$ 行为复杂的困难。以此为核心工具，作者用"最大非接吻弦集合"（maximal non-kissing collections）完整刻画了支撑 $\tau$-倾斜模（support $\tau$-tilting modules），并以组合语言显式描述了它与函子有限挠类（functorially finite torsion classes）之间的双向双射，同时给出了 $\operatorname{Ext}^1(Y,X)$ 作为短正合列的显式基底。这些结果将 McConville 在特殊组合背景下发现的现象推广到一般温和代数，为理解该类代数的倾斜理论和挠结构提供了统一的组合框架。
原文 arXiv:1707.07665;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1707.07665v2