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.