When is the Rees algebra Gorenstein?
Let \((A,m)\) be a local ring with \(\dim A \geq 1\) which has a canonical module \(K_A\), \(F\) a sequence of ideals \(I_0 = A \supseteq I_1 \supseteq I_2 \supseteq \ldots\) of \(A\) with \(I_m I_n \subseteq I_{m + n}\) for all \(m,n \geq 0\) and \(R(F) : = \bigoplus_{n \geq 0} I_n\), \(G (F) = \bigoplus_{n \geq 0} I_n/I_{n + 1}\). Then (i) If \(R(F)\) is a Cohen-Macaulay ring then there is a filtration \(\{K_n\}\) of \(K_A\) such that \(K_{R(F)} = \bigoplus_{n \geq 1} K_n\), \(K_{G(F)} = \bigoplus_{n \geq 1} K_{n - 1}/K_n\) are the canonical modules of \(R(F)\), respectively \(G(F)\). (ii) \(R(F)\) is a Gorenstein ring if and only if \(\bigoplus_{n \geq 2} [K_{G(F)}]_n \cong G (F)_{(- 2)}\) and \(R(F)\) is Cohen- Macaulay. By \textit{S. Goto} and \textit{K. Nishida} [Mem. Am. Math. Soc. 526 (1994; Zbl 0812.13016)] and \textit{Duong Quôc Viêt} [Commun. Algebra 21, No. 1, 221-229 (1993; Zbl 0772.13002)], the condition that \(R(F)\) is Cohen- Macaulay is expressed by means of the local cohomology modules of \(G(F)\). Thus (ii) characterizes the property of \(R(F)\) to be Gorenstein in terms of \(G(F)\). These results applied for the \(I\)-adic filtrations, \(I \subseteq A\), yield in particular a criterion of \textit{S. Ikeda} [Nagoya Math. J. 102, 135-154 (1986; Zbl 0585.13014)] valid also for the case grade \((I) = 1\). If grade \((I) = 1\) then there exists a Cohen-Macaulay local ring \(A\) which is not Gorenstein but \(R(I)\) is Gorenstein [there are no such examples when \(\text{grade} (I) \geq 2\) by the cited paper of Ikeda].
scientific article; zbMATH DE number 783915