On the Combinatorics of Gentle Algebras
Thomas Brüstle, Guillaume Douville, Kaveh Mousavand, Hugh Thomas, Emine Yıldırım · Canadian Journal of Mathematics · 2019
Abstract For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for $\operatorname{Hom}(X,\unicode[STIX]{x1D70F}Y)$ . We use this to describe support $\unicode[STIX]{x1D70F}$ -tilting modules for $A$ . We give a combinatorial realization of maps in both directions realizing the bijection between support $\unicode[STIX]{x1D70F}$ -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, showing that many but not all of them can be extended to general gentle algebras.