Grothendieck groups of d-exangulated categories and a modified Caldero-Chapoton map

A strong connection between cluster algebras and representation theory was established by the cluster category. Cluster characters, like the original CalderoChapoton map, are maps from certain triangulated categories to cluster algebras and they have generated much interest. Holm and Jorgensen constructed a modified Caldero-Chapoton map from a sufficiently nice triangulated category to a commutative ring, which is a generalised frieze under some conditions. In their construction, a quotient Ksp subcategory T is used. In this article, we show that this quotient is the Grothendieck group of a certain extriangulated category, thereby exposing the significance of it and the relevance of extriangulated structures. We use this to define another modified Caldero-Chapoton map that recovers the one of Holm-Jorgensen. We prove our results in a higher homological context. Suppose S is a (d + 2)angulated category with subcategories X subset of T subset of S, where X is functorially finite and T is 2d-cluster tilting, satisfying some mild conditions. We show there is an isomorphism between the Grothendieck group K0(S, EX , sX ) of the category S, equipped with the d-exangulated structure induced by X, and the quotient 0 (T)/N, where N is the higher analogue of M above. When X = T the isomorphism is induced by the higher index with respect to T introduced recently by Jorgensen. Thus, in the general case, we can understand the map taking an object in S to its K0-class in K0(S, EX , sX ) as a higher index with respect to the rigid subcategory X. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).