On the canonical module of the Rees algebra and the associated graded ring of an ideal (Q1088734)

This article continues the line of research started by \textit{J. Herzog} and \textit{W. V. Vasconcelos} [J. Algebra 93, 182--188 (1985; Zbl 0562.13003)]. It contains results on the canonical modules \(\omega\) of the Rees algebra \(S\), the extended Rees algebra \(T\) and the associated graded ring \(G\) of a local Cohen-Macaulay ring \(R\) with respect to an ideal \(I\). In the first section the authors discuss the problem under which conditions \(G\) is a Gorenstein ring. They show that \(G\) inherits the Gorenstein property from \(R\) whenever \(G\) is a domain. Furthermore: If \(I\) is generated by a d-sequence [\textit{C. Huneke}, Adv. Math. 46, 249--279 (1982; Zbl 0505.13004)] and \(R/I\) is Cohen-Macaulay, then \(G\) is Gorenstein if and only if \(I\) is strongly Cohen-Macaulay [\textit{C. Huneke}, Trans. Am. Math. Soc. 277, 739--763 (1983; Zbl 0514.13011)]. In the second section it is assumed throughout that \(S\) (hence \(G\) and \(T\)) is (are) Cohen-Macaulay. The authors say that the canonical module of \(S\) has the expected form if \(\omega_S\cong \omega_R(1,t)^m\) for some \(m\ge -1\) (where \(S=R[It]\), \(t\) an indeterminate). The following conditions are shown to be equivalent: (i) \(\omega_S\) has the expected form; (ii) \(\omega_T\cong \omega_RT\); (iii) \(\omega_G\) is the associated graded module of \(\omega_R\). It turns out that the exponent \(m\) is determined by those powers of \((1,t)\) which are Cohen-Macaulay modules, and in case \(R\) has an infinite residue field and \(I\) is an ideal primary to the maximal ideal, \(m\) can be computed explicitly: \(m=\dim(R)-\rho (I)-1\), \(\rho(I)\) denoting the reduction number of \(I\). In the last section the authors discuss conditions under which \(S\) is a Cohen-Macaulay ring. The results mainly concern the cases in which \(I\) is the maximal ideal, primary to the maximal ideal, strongly Cohen-Macaulay. They include the statement that under the pertaining hypotheses the canonical module has the expected form.