On the Combinatorics of Gentle Algebras

<jats:title>Abstract</jats:title><jats:p>For <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0008414X19000397_inline1.png" /><jats:tex-math>$A$</jats:tex-math></jats:alternatives></jats:inline-formula> a gentle algebra, and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0008414X19000397_inline2.png" /><jats:tex-math>$X$</jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0008414X19000397_inline3.png" /><jats:tex-math>$Y$</jats:tex-math></jats:alternatives></jats:inline-formula> string modules, we construct a combinatorial basis for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0008414X19000397_inline4.png" /><jats:tex-math>$\operatorname{Hom}(X,\unicode[STIX]{x1D70F}Y)$</jats:tex-math></jats:alternatives></jats:inline-formula>. We use this to describe support <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0008414X19000397_inline5.png" /><jats:tex-math>$\unicode[STIX]{x1D70F}$</jats:tex-math></jats:alternatives></jats:inline-formula>-tilting modules for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0008414X19000397_inline6.png" /><jats:tex-math>$A$</jats:tex-math></jats:alternatives></jats:inline-formula>. We give a combinatorial realization of maps in both directions realizing the bijection between support <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0008414X19000397_inline7.png" /><jats:tex-math>$\unicode[STIX]{x1D70F}$</jats:tex-math></jats:alternatives></jats:inline-formula>-tilting modules and functorially finite torsion classes. We give an explicit basis of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0008414X19000397_inline8.png" /><jats:tex-math>$\operatorname{Ext}^{1}(Y,X)$</jats:tex-math></jats:alternatives></jats:inline-formula> 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.</jats:p>