Auslander-Reiten theory for Gorenstein algebras

Question

In the paper "Cohen-Macauley and Gorenstein artin algebras", Auslander and Reiten have a short section about Auslander-Reiten theory in Gorenstein algebras (I always assume we have an artin algebra here). For Gorenstein algebras $A$ with Gorenstein dimension $g$, the category of Gorenstein projective modules has almost split sequences and the Auslander-Reiten translate of an object $C$ is given by the nonprojective part of the module $\Omega^gD\Omega^g Tr(C)$. Here are my questions:

1. Are there examples (of larger classes of Gorenstein algebras) in the literatur, where Auslander-Reiten quivers of the category of Gorenstein projectives has been calculated?

2. If there are only finitely many Gorenstein projective modules, one can build the direct sum $M$ of all such modules and calculate the endomorphism ring $B=End_A(M)$. Is there a relation between the Auslander-Reiten quiver of the Gorenstein projectives and the quiver of $B$ as is the case for "normal" Auslander algebras?

3. Are there useful tools to calculate the middle term in such an almost split sequence?

Answer 1

This is only a partial answer:

For Q1, you might be interested in the examples in the papers arXiv:math/0609138 and arXiv:1309.7301, which both give several examples of AR quivers of categories of Gorenstein projectives. The categories being considered are not described as such, particularly in the first case, but each is equivalent to the category of Gorenstein projectives over the endomorphism algebra of a projective generator; these algebras have Gorenstein dimension 1 for all the examples in both papers. I'm not sure this counts as a computation in a 'large class of examples', but they are at least some examples.

In the case of Q2, these quivers are isomorphic (or opposite, depending on various conventions), essentially by definition, except if you consider the AR-quiver to also include the data of how the AR translation acts on vertices, in which case I don't know how to recover that on the quiver of B (without using the isomorphism!), even in the case of normal Auslander algebras. This works for any additive category with AR-sequences and finitely many indecomposables (or even without the AR-sequences; if you forget the translation structure, the definition of the rest of the AR quiver makes sense without these).

However, the AR-quiver, with its translation structure, has a naturally associated mesh algebra, and this need not (quite) agree with the endomorphism algebra of a basic additive generator. Let $R=\mathbb{C}[[x^n,xy,y^n]]$ be a type A Kleinian singularity. Then $R$ is $1$-Gorenstein and Cohen–Macaulay, and the Gorenstein projectives coincide with the Cohen–Macaulay modules, finitely many of which are indecomposable. Herzog shows that a basic additive generator for the category of Gorenstein projective modules is $S=\mathbb{C}[[x,y]]$, and by a result of Reiten–van den Bergh, $\operatorname{End}_R(S)$ is isomorphic to the preprojective algebra $\Pi$ of type $\tilde{\mathsf{A}}_{n-1}$. Under this isomorphism, the indecomposable projective $R$ corresponds to the Euclidean node of the diagram, which looks exactly like all the other nodes, and indeed has a mesh relation in $\Pi$. However, the AR-translation is undefined at projective vertices, and so the mesh algebra misses out this relation. (Essentially the same thing happens in other Dynkin types as well, but I emphasize type $\mathsf{A}$ because the calculation is very tractable for small $n$.)

In these Kleinian singularity examples, the AR translation on $\operatorname{GP}(R)$ is the identity where defined, so it extends in an obvious way to the projective vertex. Making this extension gives a translation quiver whose mesh algebra is $\operatorname{End}_R(S)$, so the two aren't very far apart, but there is definitely some mismatch that might get worse for other examples.

I find this very strange (to the point where, when I first noticed it, I had to do a calculation for small $n$ to convince myself that Reiten–van den Bergh's result was really true, and the 'extra' relation does appear in $\operatorname{End}_R(S)$!) and if anyone has a good conceptual explanation of why this happens I would be very interested.

While not directly related to your question, you may also be interested in Iyama's very general version of the Auslander correspondence (arXiv:math/0411631), which includes categories of Gorenstein projective modules as a special case.

Answer 2

Concerning Q3: In QPA one can compute (although I have not tested it extensively) almost split sequences in $^\perp T = \{ X\mid\operatorname{Ext}^i_\Lambda(X,T) = 0 \textrm{ for } i >0\}$ for a cotilting module $T$. When $\Lambda$ is an admissible quotient of a path algebra and $\Lambda$ is Gorenstein, then letting $T = \Lambda$ gives you a way of computing in the category of Cohen-Macaulay modules.

I hope that this is helpful.

Best regards, the QPA-team.

Command:

AlmostSplitSequenceInPerpT( T, M );

where $T$ is the cotilting module and $M$ is an indecomposable module in $^\perp T\setminus \operatorname{add T}$.