In http://link.springer.com/chapter/10.1007%2F978-3-0348-8658-1_8#page-1 Auslander and Reiten introduced COhen-Macaulay Artin algebras as a generalisation of Gorenstein algebras. Let X be the full subcategory of modules having finite projective dimension and Y be the full subcategory of modules having finite injective dimension. An algebra A is called Cohen-Macaulay in case there is an equivalence $F:Hom(W,-): Y \rightarrow X$ for some bi-$A$-module $W$ such that $F$ is part of a pair of adjoint functor (G,F) between mod-A. 3 questions:
1.Did interesting examples of non-Gorenstein Cohen-Macaulay Artin algebras appear in the meantime?
Given a representation-finite algebra A such that X and Y have the same number of indecomposable modules up to isomorphism, is A then automatically Cohen-Macaulay? Computer tests suggest that most serial algebras have that X and Y have the same number of modules.
Is there a kind of (semi-)finite test for being Cohen-Macauley that might be implemented in a computer algebra system like QPA? For Gorensteinness one just has to calculate the minimal injective resolution of the regular module (and one might guess that it is not finite after some time)