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 τc→b→cτc→b→c be an Auslander–Reiten triangle. The map X   has the salient property that X(τc)X(c)−X(b)=1X(τc)X(c)−X(b)=1. This is part of the definition of a so-called frieze, see [1].The construction of X depends on a cluster tilting object. In a previous paper [14], we introduced a modified Caldero–Chapoton map ρ depending on a rigid object; these are more general than cluster tilting objects. The map ρ   sends objects of sufficiently nice triangulated categories to integers and has the key property that ρ(τc)ρ(c)−ρ(b)ρ(τc)ρ(c)−ρ(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 ρ.