Auslander-Gorenstein resolution.
Let \(\Lambda\) be a Noetherian ring with minimal injective resolution \(I^\bullet\). In the paper under review, \(\Lambda\) is said to satisfy the Auslander Condition if for every \(i\) one has \(\text{flat.dim}(I^i)\leq i\), and \(\Lambda\) is said to be an Auslander-Gorenstein ring if, in addition, \(\text{id}_\Lambda(\Lambda)=\text{id}_{\Lambda^{\text{op}}}(\Lambda)<\infty\). Auslander-Gorenstein rings, which have also been called \(m\)-Gorenstein rings in the literature, are a generalization of commutative Gorenstein rings. These arise naturally in various settings as described in the introduction of the paper under review. In this paper, the authors introduce the notion of Auslander-Gorenstein resolutions. The main theorem states that a ring is Auslander-Gorenstein if and only if it admits an Auslander-Gorenstein resolution over another Auslander-Gorenstein ring. Despite their prevalence, there are few known methods for constructing Auslander-Gorenstein rings. The main result of this paper may be viewed as providing one such construction. The utility of this method is illustrated by several examples in which Auslander-Gorenstein resolutions are explicitly computed.
scientific article; zbMATH DE number 5999994
scientific article