Minimal injective resolutions of Cohen-Macaulay isolated singularities
This paper studies (not necessarily commutative) finitely generated algebras \(\Lambda\) over a Gorenstein complete local ring \((R, {\mathfrak m})\) which are Cohen-Macaulay isolated singularities and finitely generated \(\Lambda\)-modules \(M\) which are Cohen-Macaulay \(R\)-modules with \(\dim_R M = \dim R\). The main result is a description of the minimal injective resolution of \(M\) in terms of the Cousin complex of \(M\) (as an \(R\)-module) and the minimal projective resolution of the \(R\)-dual of \(M\). As a consequence, the authors prove that if \(R\) is a regular local ring, \(\Lambda\) is \(k\)-Gorenstein if and only if the ring \((R/{\mathfrak m})\otimes_R\Lambda\) is \((k-\dim R)\)-Gorenstein. These results generalize earlier results of \textit{H. Fujita} [Commun. Algebra 26, No. 2, 447-451 (1998; Zbl 0895.16006)]. The paper concludes with several examples of Cohen-Macaulay isolated singularities.
scientific article; zbMATH DE number 1394462