Generic bases of skew-symmetrizable affine type cluster algebras

Geiss, Leclerc and Schröer introduced a class of 1-Iwanaga–Gorenstein algebras H associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. They also proved that indecomposable rigid H-modules of finite projective dimension are in bijection with non-initial cluster variables of the corresponding Fomin–Zelevinsky cluster algebra. In this article, we prove in all affine types that their conjectural Caldero–Chapoton type formula on these modules coincide with the Laurent expression of cluster variables. By taking generic Caldero–Chapoton functions on varieties of modules of finite projective dimension, we obtain bases for affine type cluster algebras with full-rank coefficients containing all cluster monomials.