The Grorthendieck group of an extriangulated category

In this paper, we investigate the split Grothendieck group Kˢᵖ₀(M) of a d-rigid subcategory M in an extriangulated category C. As applications, we prove the following results: (1) If M is a silting subcategory, then the Grothendieck group K₀(C) is isomorphic to K₀ˢᵖ(M); (2) If M is a d-cluster tilting subcategory, then K₀(C) is isomorphic to the index Grothendieck group K₀ⁱⁿ(M); (3) Let C_Aₙᵈ be the d-cluster category of type Aₙ. If d is even, then K₀(C_Aₙᵈ) Z/(n+1)Z. If d is odd, then K₀(C_Aₙᵈ) Z if n is odd; K₀(C_Aₙᵈ) 0 if n is even.