The Grothendieck group of a triangulated category

TL;DR

This paper provides a direct proof that the Grothendieck group of a triangulated category with a silting subcategory is isomorphic to the split Grothendieck group of that subcategory, also extending to cluster-tilting cases.

Contribution

It offers a direct proof of a known isomorphism and introduces its analogue in the cluster-tilting context.

Findings

Grothendieck group is isomorphic to split Grothendieck group with silting subcategory

Extension of results to cluster-tilting analogue

Provides a more direct proof of a known theorem

Abstract

We give a direct proof of the following known result: the Grothendieck group of a triangulated category with a silting subcategory is isomorphic to the split Grothendieck group of the silting subcategory. Moreover, we obtain its cluster-tilting analogue.