Krull--Gabriel dimension and Cohen-Macaulay representations

Abstract

Let $R$ be a complete Cohen--Macaulay local ring and $\Lambda$ an $R$-order. We study the Krull--Gabriel dimension of the functor category $\mathrm{mod}\,\underline{\mathcal C}(\Lambda)$, where $\underline{\mathcal C}(\Lambda)$ is the stable category of maximal Cohen--Macaulay $\Lambda$-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 $\Lambda$ is Gorenstein and $\mathrm{KGdim}\,\mathrm{mod}\,\underline{\mathcal C}(\Lambda)\leq 1$, then $\Lambda$ is an isolated singularity. We then show that, if $\Lambda$ is an isolated singularity of uncountable Cohen--Macaulay representation type, then $\mathrm{KGdim}\,\mathrm{mod}\,\underline{\mathcal C}(\Lambda)\neq 1$.