On multi-Rees algebras. With an appendix by Ngô Viêt Trung
Let \(I\) denote an ideal of positive grade in a local ring \((A,{\mathfrak m})\). The main aim of the paper is two-fold. Firstly the authors investigate the Cohen-Macaulayness and the Gorensteinness of the multigraded Rees ring \(R_ A (I; k_ 1, \dots, k_ r) : = A[I^{k_ 1} t_ 1, \dots, I^{k_ r} t_ r]\), where \(t_ 1, \dots, t_ r\) are independent variables over \(A\). The Cohen-Macaulay property of \(R_ A (I; k_ 1, \dots, k_ r)\) is determined by a vanishing behaviour of the local cohomology of \(R_ A (I) : = A[It]\), the ordinary Rees ring of \(A\) with respect to \(I\). In a second main result it is shown that \(R_ A (I; k_ 1, \dots, k_ r)\) is a Gorenstein ring if and only if \(R_ A (I^{k_ 1 + \cdots + k_ r})\) is a Gorenstein ring. It is noteworthy to say that there are no Cohen-Macaulay assumptions on \(R_ A (I)\) or \(A\) for the proof of these results. This leads the authors to the second main theme of the paper: To describe the Gorenstein property of \(R_ A (I^ q)\) for an integer \(q \in \mathbb{N}_ +\). Under the additional hypothesis of \(\text{grade}_ AI > 1\) and a weak vanishing result of the local cohomology of \(R_ A (I)\) it is shown that \(R_ A (I^ q)\) is a Gorenstein ring if and only if (i) \(R_ A (I^ q)\) is a Cohen-Macaulay ring, (ii) \(A\) is a quasi-Gorenstein ring, and (iii) the canonical module of the form ring of \(A\) with respect to \(I\) is isomorphic to gr\(_ A^*(I) (- q - 1)\), where gr\(_ A^* (I)\) denotes the form ring of \(A\) with respect to the Ratliff-Rush closures of \(I^ n\), \(n \in \mathbb{N}\). -- There are further results omitting the additional statements on the local cohomology and the grade. The main technical tools are: (1) a theory of multigraded rings and modules started by \textit{E. Hyry} [Manuscr. Math. 80, No. 1, 13-20 (1993; Zbl 0812.13005)]; (2) the use of Veronesean rings and Segre products; (3) investigations about the multigraded local cohomology modules and the canonical moduli of multigraded Rees rings. In the appendix (written by \textit{Ngô Viêt Trung}) there are examples, in particular those where \(R_ A (I;1,1)\) is a Gorenstein ring while \(R_ A (I)\) is not even a Cohen-Macaulay ring.
scientific article; zbMATH DE number 734571
scientific article