For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for 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 Ext$^1(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.

中文翻译：

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关于温和代数的组合

对于 $A$ 一个温和的代数，以及 $X$ 和 $Y$ 字符串模块，我们为 Hom($X,\tau Y$) 构建了一个组合基础。我们用它来描述对 $A$ 的支持 $\tau$-tilting 模块。我们在两个方向上给出了映射的组合实现，实现了支持 $\tau$-tilting 模块和功能有限扭转类之间的双射。我们给出了 Ext$^1(Y,X)$ 作为短精确序列的明确基础。我们分析了 McConville 在更受限制的组合环境中给出的几种构造，表明它们中的许多但不是全部都可以扩展到一般的温和代数。