We generalize the results of Kahn about a correspondence between Cohen-Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen-Macaulay finite, but are Cohen-Macaulay tame.

Given a semiperfect two-sided noetherian ring Λ, we study two subcategories ${}_k\left(\Lambda \right)=M\in mod\Lambda |Ex{t}_{\Lambda }^j(TrM,\Lambda )=0\left(1\le j\le k\right)$ and ${}_k\left(\Lambda \right)=N\in mod\Lambda |Ex{t}_{\Lambda }^j(N,\Lambda )=0\left(1\le j\le k\right)$ of the category mod Λ of finitely generated right Λ-modules, where Tr M is Auslander’s transpose of M. In particular, we give another convenient description of the categories ${}_k\left(\Lambda \right)$ and ${}_k\left(\Lambda \right)$, and we study category equivalences and stable equivalences between them. Several results proved in [J. Algebra 301 (2006), 748-780] are extended to the case when Λ is a two-sided noetherian semiperfect ring.