Gabriel-Roiter measure, representation dimension and rejective chains

Abstract

The Gabriel-Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel-Roiter measures. Using this notion, we prove the following broader statement: given any object X and any Gabriel-Roiter measure µ in an abelian length category A, there exists an object X ′ which depends on X and µ, such that Γ = End A (X ⊕ X ′ ) has finite global dimension. Analogously to Iyama's original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.