Auslander-Reiten's Cohen-Macaulay algebras and contracted preprojective algebras

Abstract

Auslander and Reiten called a finite dimensional algebra $A$ over a field Cohen-Macaulay if there is an $A$-bimodule $W$ which gives an equivalence between the category of finitely generated $A$-modules of finite projective dimension and the category of finitely generated $A$-modules of finite injective dimension. For example, Iwanaga-Gorenstein algebras and algebras with finitistic dimension zero on both sides are Cohen-Macaulay, and tensor products of Cohen-Macaulay algebras are again Cohen-Macaulay. They seem to be all of the known examples of Cohen-Macaulay algebras. In this paper, we give the first non-trivial class of Cohen-Macaulay algebras by showing that all contracted preprojective algebras of Dynkin type are Cohen-Macaulay. As a consequence, for each simple singularity $R$ and a maximal Cohen-Macaulay $R$-module $M$, the stable endomorphism algebra ${\underline{End}}_R(M)$ is Cohen-Macaulay. We also give a negative answer to a question of Auslander-Reiten asking whether the category $CMA$ of Cohen-Macaulay $A$-modules coincides with the category of $d$-th syzygies, where $d\ge 1$ is the injective dimension of $W$. In fact, if $A$ is a Cohen-Macaulay algebra that is additionally $d$-Gorenstein in the sense of Auslander, then $CMA$ always coincides with the category of $d$-th syzygies.