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 |$X^{\prime}$| that depends on |$X$| and |$\mu$|⁠, such that |$\Gamma =\operatorname{End}_{\mathcal{A}}(X\oplus X^{\prime})$| has finite global dimension. Analogously to Iyama’s original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.

中文翻译：

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Gabriel-Roiter测度，表示维度和排斥链

由Iyama在2002年建立的结果，使用Gabriel–Roiter测度提供了Artin代数表示维有限性的另一种证明。Gabriel–Roiter测度的概念可以扩展到阿贝尔长度类别，每个此类类别多种加百利-罗伊特度量。使用这个概念，我们证明以下更广泛的陈述：给定任何对象| $ X $ | 以及任何加百利–罗伊特度量| $ \ mu $ | 在阿贝尔长度类别| $ \ mathcal {A} $ |⁠中，存在一个对象| $ X ^ {\ prime} $ | 取决于| $ X $ | 和| $ \ mu $ |⁠，使得| $ \ Gamma = \ operatorname {End} _ {\ mathcal {A}}（X \ oplus X ^ {\ prime}）$ |具有有限的全局范围。与Iyama的原始结果类似，我们的构造产生准遗传环，并适合于排斥链理论。