# Cohen-Macaulay Artin algebras

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 $$F:Hom_A(W,-): Y \rightarrow X$$ for some bi-$$A$$-module $$W$$ such that $$F$$ is part of a pair of adjoint functor (G,F) between mod-A. In case one can even choose $$W=A$$, the algebra is called Gorenstein. For algebras with left and right finitistic dimension zero one can choose $$W=D(A)$$.

3 questions:

1.Did interesting examples of non-Gorenstein Cohen-Macaulay Artin algebras appear in the meantime? In the article of Auslander and Reiten there seems to be no non-trivial examples except for algebras with left and right finitistic dimension zero (for example local algebras) and certain tensor products. Besides that, no non-trivial examples seem to be known or appeared during the last 30 years.

1. Is there a kind of 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)

2. A necessary condition is that that the strong cotilting module $$X$$ (that is the unique basic cotilting module $$X$$ such that $$Ext_A^1(X,U)=0$$ for all $$U$$ of finite injective dimension) has the property that $$End_A(X) \cong A$$. Curiously that seems to never hold for most algebras. For example I have the following

Guess: A Nakayama algebra is a Cohen-Macaulay Artin algebra iff it is Gorenstein. It seems that $$End_A(X)$$ never has the same dimension as $$A$$.

edit: It seems a necessary and sufficient condition for an algebra $$A$$ to be Cohen-Macaulay is that the strong cotilting module $$X$$ has $$End_A(X) \cong A$$ and every simple module is a sumand of $$top(X)$$, see section 6 of https://www.sciencedirect.com/science/article/pii/0001870891900378 . So a finite test is possible for representation finite algebras.

• All commutative Artin rings are Cohen-Macauay, very few are Gorenstein. – Mohan Jan 14 '17 at 3:52
• Yes, they are Gorenstein iff they are selfinjective. – Mare Jan 14 '17 at 9:46