The Grothendieck group of an extriangulated category

TL;DR

This paper studies the split Grothendieck group of a subcategory within an extriangulated category, establishing isomorphisms with other Grothendieck groups under certain conditions and computing specific examples.

Contribution

It provides new isomorphism results relating the Grothendieck group of an extriangulated category to those of its subcategories, including explicit calculations for $A_n$-type cluster categories.

Findings

$K_0( ext{category})$ is isomorphic to $K_0^{ m sp}( ext{subcategory})$ for silting subcategories.

$K_0( ext{category})$ is isomorphic to $K_0^{ m in}( ext{subcategory})$ for $d$-cluster tilting subcategories.

Explicit computation of $K_0$ for $d$-cluster categories of type $A_n$, depending on parity of $d$ and $n$.

Abstract

In this paper, we investigate the split Grothendieck group $K^{\rm sp}_{0}(\mathcal{M})$ of a $d$-rigid subcategory $\mathcal{M}$ in an extriangulated category $\mathscr{C}$. As applications, we prove the following results: (1) If $\mathcal{M}$ is a silting subcategory, then the Grothendieck group $K_{0}(\mathscr{C})$ is isomorphic to $K_{0}^{\rm sp}(\mathcal{M})$; (2) If $\mathcal{M}$ is a $d$-cluster tilting subcategory, then $K_{0}(\mathscr{C})$ is isomorphic to the index Grothendieck group $K_{0}^{\rm in}(\mathcal{M})$; (3) Let $\mathcal{C}_{A_{n}}^{d}$ be the $d$-cluster category of type $A_n$. If $d$ is even, then $K_0(\mathcal{C}_{A_{n}}^{d})\cong \mathbb{Z}/(n+1)\mathbb{Z}$. If $d$ is odd, then $K_0(\mathcal{C}_{A_{n}}^{d})\cong \mathbb{Z}$ if $n$ is odd; $K_0(\mathcal{C}_{A_{n}}^{d})\cong 0$ if $n$ is even.