Towards a theory of Bass numbers with application to Gorenstein algebras. (Q2773347)

A commutative Noetherian Gorenstein ring \(R\) is usually defined by the property: \(R_{\mathbf p}\) has finite injective dimension for all \({\mathbf p}\in\text{Spec\,}R\). NEWLINENEWLINEThere have been several ways to define Gorenstein noncommutative algebras over a commutative Noetherian ring \(R\). This paper defines a noncommutative module-finite \(R\)-algebra \(A\) to be Gorenstein if the Cousin complex of \(A\) is a minimal injective resolution. The notion of a Cousin complex was developed by \textit{R. Y. Sharp} [in Math. Z. 112, 340--356 (1969; Zbl 0182.06103)], where it was used to characterize commutative Gorenstein rings. The new definition in the noncommutative case has the advantage of carrying over almost all of the commutative theory.NEWLINENEWLINEFor instance, \(A\) is Gorenstein if and only if \(A\) is a Cohen-Macaulay ring with \(\text{id}_{A_{\mathbf p}}=\text{Kdim}_{R_{\mathbf p}}A_{\mathbf p}\) for all \({\mathbf p}\in\text{Supp}_RA\), where \(\text{id}_{A_{\mathbf p}}\) denotes the injective dimension and \(\text{Kdim}_{R_{\mathbf p}}A_{\mathbf p}\) the Krull dimension of \(A_{\mathbf p}\). The authors show that the commutative theory of Bass numbers can be developed for the noncommutative case with no extra changes. In particular, Gorensteinness can be characterized in terms of Bass numbers.NEWLINENEWLINEIf \(R\) is a local ring, every Gorenstein \(R\)-algebra is its own canonical module and the Gorensteinness is inherited under flat base changes. Moreover, if \(A\) is a local ring, they show that Gorensteinness can be characterized by several conditions similar to those in the commutative case.NEWLINENEWLINEThe paper is a rich source for the study of noncommutative Gorenstein rings. For further developments see two other papers of the authors [Arch. Math. 73, No. 4, 249--255 (1999; Zbl 0967.13020) and J. Lond. Math. Soc., II. Ser. 63, No. 2, 319--335 (2001; Zbl 1047.13003)].