On the global structure of regular orders of dimension two

Let O be a regular domain of dimension two with field of fractions K. λ is an O-order in a separable K-algebra A. λ is said to be endo-regular (semi-endo-regular), if End λ ;(M) has global dimension two for every finitely generated (indecomposable) Cohen Macaulay-module M. We first show that these conditions are inherited by the localizations and completions at the maximal ideals in O. For endo-regular orders the converse also holds, and we give a complete description of them. For the semi-endo-regular orders we give a description in the complete situation. Globally we give examples, based on algebraic geometry, which show that the converse implications are not true.