A Gordon--Green-type result for Gorenstein-projective modules

Let $\mathrm{\Lambda }$ be a $\mathbb{Z}$-graded Artin algebra which is Iwanaga–Gorenstein. We prove that $\mathrm{\Lambda }$ is CM-finite if and only if the category of gradable Gorenstein-projective modules has finite type, and if and only if the lengths of the indecomposable gradable Gorenstein-projective modules are bounded. Moreover, if $\mathrm{\Lambda }$ is CM-finite, then every Gorenstein-projective $\mathrm{\Lambda }$-module is gradable. These results provide Gorenstein-projective analogues of the corresponding theorems of Gordon–Green for graded Artin algebras of finite representation type.