Extensions in Jacobian Algebras and Cluster Categories of Marked Surfaces

In the context of representation theory of finite dimensional algebras, string algebras have been extensively studied and almost all aspects of their representation theory are well-understood. One exception to this is the classification of extensions between indecomposable modules. In this paper we explicitly describe such extensions for a class of string algebras, namely gentle algebras associated to surface triangulations. These algebras arise as Jacobian algebras of unpunctured surfaces. We give bases of their extension spaces and show that the dimensions of these extension spaces are given in terms of crossing arcs in the surface. Our approach is new and consists of interpreting snake graphs as indecomposable modules. To give a complete answer, we need to work in the associated cluster category where we explicitly calculate the middle terms of extensions and give a basis of the extension space. We note that not all extensions in the cluster category give rise to extensions for the Jacobian algebra.