# Gabriel-Roiter measure, representation dimension and rejective chains

## TL;DR

This paper extends the Gabriel-Roiter measure to abelian length categories, providing new proofs of finiteness results for representation dimension and connecting to rejective chains and quasihereditary rings.

## Contribution

It generalizes the Gabriel-Roiter measure to broader categories and demonstrates its use in establishing finite global dimension and related structures.

## Key findings

- Existence of objects with finite global dimension endomorphism rings
- Extension of Gabriel-Roiter measure to abelian length categories
- Connection to rejective chains and quasihereditary rings

## 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 $\mu$ in an abelian length category $\mathcal{A}$, there exists an object $X'$ which depends on $X$ and $\mu$, such that $\Gamma = \operatorname{End}_{\mathcal{A}}(X \oplus X')$ has finite global dimension. Analogously to Iyama's original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.