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Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings.As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: ✩ Part of this work was done at MSRI during the spring semester of 2003, when the authors participated in the Program in Commutative Algebra. We thank the institution and program organizers for a very stimulating research environment.

Abstract. Let R be a commutative Noetherian ring. We study R-modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count among them such, potentially, diverse objects as dualizing complexes for R on one side, and on the other, the ring itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes "lying between" these extremes is incentive.

The classical homological dimensions-the projective, flat, and injective onesare usually defined in terms of resolutions and then proved to be computable in terms of vanishing of appropriate derived functors. In this paper we define restricted homological dimensions in terms of vanishing of the same derived functors but over classes of test modules that are restricted to assure automatic finiteness over commutative Noetherian rings of finite Krull dimension. When the ring is local, we use a mixture of methods from classical commutative algebra and the theory of homological dimensions to show that vanishing of these functors reveals that the underlying ring is a Cohen-Macaulay ring-or at least close to being one.  2002 Elsevier Science (USA)

Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture-by a 2003 counterexample due to Jorgensen andŞega-motivates the consideration of the class of rings that do satisfy Auslander's condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein. Furthermore, the Auslander-Reiten Conjecture is proved for AC rings, and Auslander's G-dimension is shown to be functorial for AC rings that are commutative or have a dualizing complex.