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

Abstract

The investigations of this paper originate in the following question: Suppose I is an ideal of a local Gorenstein ring (R, m), whose associated graded ring gr,(R) is CohenMacaulay. Under which extra conditions is gr,(R) actually a Gorenstein ring? Hochster shows in [ 151 that gr,(R) is a Gorenstein ring if R is factorial and gr,(R) is a domain. Hochster’s arguments work as well if one only requires R to be a Gorenstein ring. However, the condition that gr,(R) is a domain cannot be weakened much. In fact, we given an example of a local complete intersection (R, m) whose associated graded ring gr,,,(R) is a reduced Cohen-Macaulay ring, but not a Gorenstein ring. The question when gr,(R) is a Gorenstein ring has a more satisfying answer for ideals I generated by a d-sequence, for which R/I is a Cohen-Macaulay ring. d-sequences were introduced and studied by Huneke. Their relevance in the study of powers of ideals was shown by Huneke in [ 161. For further investigations on d-sequences the reader is referred to [ 111. Now, given an ideal I generated by a d-sequence and such that R/I is CohenMacaulay we show that gr,(R) is a Gorenstein ring if and only if I is strongly Cohen-Macaulay. Again, the notion “strongly Cohen-Macaulay” was introduced by Huneke. It means that the Koszul homology of a system of generators of I is Cohen-Macaulay. Huneke showed in [ 191 that any ideal in the linkage class of a complete intersection is strongly Cohen-Macaulay. This result provides us with plenty of interesting examples. The main subject of the paper, however, is the study of the canonical module of gr,(R) and of the Rees algebra R[If] = R@Zr@12t2@ ... sR[t]. In [14] the canonical class [os] of the Rees algebra S= R[Zr] was determined under the assumption that I is