Generic bases of skew-symmetrisable affine type cluster algebras

Geiss, Leclerc and Schroeer introduced a class of 1-Iwanaga--Gorenstein algebras $H$ associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. Indecomposable rigid $H$-modules of finite projective dimension are known in bijection with non-initial cluster variables of the corresponding Fomin--Zelevinsky cluster algebra. We prove in all affine types that a Caldero--Chapoton type function on these modules proposed by Geiss--Leclerc--Schr\"oer coincide with 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.