The Grothendieck group of an extriangulated category

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.

Figures (8)

• Figure 1: The Auslander quiver of $\mathcal{C}_{A_{n}}^{d}$ when $d$ is odd.
• Figure 2: The Auslander quiver of $\mathcal{C}_{A_{n}}^{d}$ when $d$ is even.
• Figure 3: The $(d+2)$-angulation $T=\oplus_{i=1}^{n}T_i$.
• Figure 4: The Auslander-Reiten triangle $\Sigma T_1\longrightarrow T_2\longrightarrow X\longrightarrow \Sigma^{2} T_1$.
• Figure 5: The Auslander-Reiten triangle $\Sigma^{d-1} T_i\longrightarrow T_{i-1}\oplus T_{i+1}\longrightarrow \Sigma^{-1}T_i\longrightarrow\Sigma^{d} T_i$.

Theorems & Definitions (53)

• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Definition 2.1
• Proposition 2.2
• Remark 2.3
• Definition 3.1
• Example 3.2
• Definition 3.3