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

TL;DR

This paper characterizes when n-cluster tilting subcategories of module categories over artin algebras have additive generators, linking this to the basis of n-almost split sequences for Grothendieck groups and properties of effaceable functors.

Contribution

It establishes a precise equivalence between the existence of additive generators in n-cluster tilting subcategories and the basis property of n-almost split sequences for Grothendieck groups.

Findings

Additive generator existence is equivalent to n-almost split sequences forming a basis.

Finite type n-cluster tilting subcategories imply basis property for Grothendieck groups.

Effaceable functors have finite length under these conditions.

Abstract

Let $\Lambda$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of mod$\Lambda$. We show that $\mathcal{M}$ has an additive generator if and only if the n-almost split sequences form a basis for the relations for the Grothendieck group of $\mathcal{M}$ if and only if every effaceable functor $\mathcal{M}\rightarrow Ab$ has finite length. As a consequence we show that if mod$\Lambda$ 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 $\Lambda$.