Cohen-Macaulay isolated singularities with a dualizing module.
Let \(R\) be a complete local Gorenstein domain of dimension \(d\), and let \(\Lambda\) be a Cohen-Macaulay \(R\)-order which is an isolated singularity in the sense of \textit{M. Auslander} [Lect. Notes Math. 1178, 194-242 (1986; Zbl 0633.13007)], i.e. \(\text{gld\,}\Lambda_p=\dim R_p\) for non-maximal primes \(p\). By \(\Lambda\mathbf{CM}\) we denote the category of (maximal) Cohen-Macaulay representations of \(\Lambda\). The author continues his former study (see [Algebr. Represent. Theory 5, No. 2, 137-147 (2002; Zbl 1003.16005)], for the one-dimensional case) of dualizing \((\Lambda,\Lambda)\)-bimodules, generalizing previous results of \textit{H.-B. Foxby} [Math. Scand. 31(1972), 267-284 (1973; Zbl 0272.13009)] to the non-commutative case. By definition, a bimodule \(\omega\in\Lambda\mathbf{CM}\) is called dualizing if it is balanced, (left and right) self-orthogonal with finite injective dimension, such that \(\text{Ext}^d_\Lambda(S,\Hom_R(\omega,R))\not=0\) for simple \(\Lambda\)-modules \(S\). Thus dualizing \(\Lambda\)-modules are fairly similar to cotilting modules. Following Foxby, the author associates two subcategories \(\mathcal A\) and \(\mathcal B\) of modules to any dualizing module \(\omega\) which are equivalent via the Hom- and tensor functor of \(\omega\). Furthermore, he shows that the \(\Lambda\)-modules of finite Gorenstein-dimension belong to \(\mathcal A\). The two categories \(\mathcal A\) and \(\mathcal B\) coincide if and only if \(\Lambda\) is a Gorenstein order. Apart from this criterion, the author provides eight other (essentially asymmetric) characterizations of this property. The results are illustrated by examples related to almost Bass orders (introduced by Hijikata in his study of local orders) and \(k\)-Gorenstein algebras.
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