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 $\mu$ 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.

The Quarterly Journal of Mathematics, 71, no. 2, pp. 619-635