The canonical module of an associated graded ring

Abstract

at) module of the Rees ring S = G I"t c R[t] of a commutative noetherian Ring R with n=O respect to an ideal I. We want to complement their results by determining the canonical module of the corresponding associated graded ring S/SI ~ ~ P/I" + 1 from the canonin=0 cal module of S. (For the theory of the canonical module we refer to [3].) Theorem. Let R be a Cohen-Macaulay ring (locally, always) 1 c R an ideal of height at least 2, S the Rees ring of R with respect to I, and G = S/SI the associated graded ring. Assume that S and G are Cohen-Macaulay rings, and that S has a canonical module cos. Then G has a canonical module r and: (i) If co s can be embedded into S such that cos (considered as an ideal now) is not contained in a minimal prime ideal of SI or Sit, then co~ ~- (cos + SI)/S1. (ii) Such an embedding exists if and only if the localizations Se with respect to the prime ideals P ~ S minimal over O, SI or Sit are Gorenstein rings. Before embarking on the proof one should note that under the Cohen-Macaulay hypothesis on R, S, and G the ideals SI and Sit are unmixed of height 1. P r o o f. By virtue of [3], Satz 5.12 the G-module Ext,(G, cos) is canonical for G. Since the adjunction of a new indeterminate to R does not affect the validity of (i), we may assume in its proof that the residue class rings R/R n P for the prime ideals mentioned and R/Q for the minimal prime ideals Q c R of I and 0 have infinitely many elements. Let us write co for cos c R. Then the following holds: