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Gabriel-Roiter measure, representation dimension and rejective chains

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