Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object, II

It is an important aspect of cluster theory that cluster categories are "categorifications" of cluster algebras. This is expressed formally by the (original) Caldero-Chapoton map X which sends certain objects of cluster categories to elements of cluster algebras. Let \tau c --> b --> c be an Auslander-Reiten triangle. The map X has the salient property that X(\tau c)X(c) - X(b) = 1. This is part of the definition of a so-called frieze. The construction of X depends on a cluster tilting object. In a previous paper, we introduced a modified Caldero-Chapoton map \rho depending on a rigid object; these are more general than cluster tilting objects. The map \rho sends objects of sufficiently nice triangulated categories to integers and has the key property that \rho(\tau c)\rho(c) - \rho(b) is 0 or 1. This is part of the definition of what we call a generalised frieze. Here we develop the theory further by constructing a modified Caldero-Chapoton map, still depending on a rigid object, which sends objects of sufficiently nice triangulated categories to elements of a commutative ring A. We derive conditions under which the map is a generalised frieze, and show how the conditions can be satisfied if A is a Laurent polynomial ring over the integers. The new map is a proper generalisation of the maps X and \rho.