The canonical module of an associated graded ring
Herzog, Simis and Vasconcelos have computed, in certain circumstances, the canonical module of the Rees ring \(S=\oplus^{\infty}_{n=0}I^ nt\subseteq R[t]\) of a commutative Noetherian ring R with respect to an ideal I. This paper is concerned with the associated graded ring \(G=S/SI\cong \oplus^{\infty}_{n=0}I^ n/I^{n+1}.\quad Suppose\) that R is Cohen-Macaulay and the ideal I of R has height at least 2; suppose also that S and G are both Cohen-Macaulay rings and that S has a canonical module \(\omega_ S\). The main result of the paper states that G has a canonical module \(\omega_ G\), and if \(\omega_ S\) can be embedded into S in such a way that \(\omega_ S\) (considered as an ideal) is not contained in a minimal prime ideal of SI or SIt, then \(\omega_ G\cong (\omega_ S+SI)/SI\); furthermore, such an embedding exists if and only if all the localizations \(S_ p\) at prime ideals P of S which are minimal over O, SI or SIt are Gorenstein rings.
scientific article; zbMATH DE number 4021295
scientific article