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

Abstract

A strong connection between cluster algebras and representation theory was established by the cluster category. Cluster characters, like the original Caldero-Chapoton (CC) map, are maps from certain triangulated categories to cluster algebras and they have generated much interest. Holm and J{\o}rgensen constructed a modified CC map from a sufficiently nice triangulated category to a commutative ring, which is a generalised frieze under some conditions. In their construction, a quotient $K_{0}^{sp}(\mathcal{T})/M$ of a Grothendieck group of a cluster tilting subcategory $\mathcal{T}$ 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 CC map that recovers the one of Holm--J{\o}rgensen. We prove our results in a higher homological context. Suppose $\mathcal{S}$ is a $(d+2)$-angulated category with subcategories $\mathcal{X}\subseteq\mathcal{T}\subseteq\mathcal{S}$, where $\mathcal{X}$ is functorially finite and $\mathcal{T}$ is $2d$-cluster tilting, satisfying some mild conditions. We show there is an isomorphism between the Grothendieck group $K_{0}(\mathcal{S},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$ of the category $\mathcal{S}$, equipped with the $d$-exangulated structure induced by $\mathcal{X}$, and the quotient $K_{0}^{sp}(\mathcal{T})/N$, where $N$ is the higher analogue of $M$ above. When $\mathcal{X}=\mathcal{T}$ the isomorphism is induced by the higher index with respect to $\mathcal{T}$ introduced recently by J{\o}rgensen. Thus, in the general case, we can understand the map taking an object in $\mathcal{S}$ to its $K_{0}$-class in $K_{0}(\mathcal{S},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$ as a higher index with respect to the rigid subcategory $\mathcal{X}$.