Relations for Grothendieck groups of $n$-cluster tilting subcategories

Abstract

Let Λ be an artin algebra and M be an n-cluster tilting subcategory of mod Λ. We show that M has an additive generator if and only if the n-almost split sequences form a basis for the relations for the Grothendieck group of M if and only if every effaceable functor M → Ab has finite length. As a consequence we show that if mod Λ has n-cluster tilting subcategory of finite type then the n-almost split sequences form a basis for the relations for the Grothendieck group of Λ.