Generic Bases for Cluster Algebras From the Cluster Category

Inspired by recent work of Geiss–Leclerc–Schröer, we use Hom-finite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by the cluster character on objects having the same index. If the matrix associated to the quiver is of full rank, then we prove that the elements in this set are linearly independent. If the cluster algebra arises from the setting of Geiss–Leclerc–Schröer, then we obtain the basis found by these authors. We show how our point of view agrees with the spirit of conjectures of Fock–Goncharov concerning the parametrization of a basis of the upper cluster algebra by points in the tropical $\mathcal X$-variety.