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A frieze in the modern sense is a map from the set of objects of a triangulated category C to some ring. A frieze X is characterised by the property that if τx→y→x is an Auslander-Reiten triangle in C, then X(τx)X(x)-X(y)=1. The canonical example of a frieze is the (original) Caldero-Chapoton map, which send objects of cluster categories to elements of cluster algebras. Holm and Jørgensen (Nagoya Math J 218:101-124, 2015; Bull Sci Math 140:112-131, 2016), the notion of generalised friezes is introduced. A generalised frieze X′ has the more general property that X′(τx)X′(x)-X′(y)∈{0,1}. The canonical example of a generalised frieze is the modified Caldero-Chapoton map, also introduced in Holm and Jørgensen (2015, 2016). Here, we develop and add to the results in Holm and Jørgensen (2016). We define Condition F for two maps α and β in the modified Caldero-Chapoton map, and in the case when C is 2-Calabi-Yau, we show that it is sufficient to replace a more technical “frieze-like” condition from Holm and Jørgensen (2016). We also prove a multiplication formula for the modified Caldero-Chapoton map, which significantly simplifies its computation in practice. [ABSTRACT FROM AUTHOR]

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 τ c → b → c be an Auslander–Reiten triangle. The map X has the salient property that X ( τ c ) 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 ( τ c ) 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 . [ABSTRACT FROM AUTHOR]