AI 中文总结
本文针对R-序Λ的极大Cohen-Macaulay模稳定范畴的函子范畴，研究其Krull–Gabriel维数，结合Herzog与Krause的不存在定理，证明了Gorenstein序及不可数表示型孤立奇点序的相关维数性质。
AI 中文摘要
设R为完备Cohen-Macaulay局部环，Λ为R-序。我们研究函子范畴mod 𝒞̲(Λ)的Krull–Gabriel维数，其中𝒞̲(Λ)是极大Cohen-Macaulay Λ-模的稳定范畴。本研究由Herzog与Krause关于Artin代数上Krull–Gabriel维数为1的不存在定理驱动。我们首先证明：若Λ为Gorenstein且KGdim mod 𝒞̲(Λ)≤1，则Λ为孤立奇点；随后证明：若Λ为具有不可数Cohen-Macaulay表示型的孤立奇点，则KGdim mod 𝒞̲(Λ)≠1。
英文摘要
Let $R$ be a complete Cohen--Macaulay local ring and $Λ$ an $R$-order. We study the Krull--Gabriel dimension of the functor category $\mathrm{mod}\,\underline{\mathcal C}(Λ)$, where $\underline{\mathcal C}(Λ)$ is the stable category of maximal Cohen--Macaulay $Λ$-modules. This work is motivated by the non-existence theorem of Herzog and Krause for Krull--Gabriel dimension $1$ over artin algebras. We first prove that, if $Λ$ is Gorenstein and $\mathrm{KGdim}\,\mathrm{mod}\,\underline{\mathcal C}(Λ)\leq 1$, then $Λ$ is an isolated singularity. We then show that, if $Λ$ is an isolated singularity of uncountable Cohen--Macaulay representation type, then $\mathrm{KGdim}\,\mathrm{mod}\,\underline{\mathcal C}(Λ)\neq 1$.
Comments19 pages
