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On the canonical module of the Rees algebra and the associated graded ring of an ideal. (English) Zbl 0613.13007
This article continues the line of research started by J. Herzog and 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 [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 [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 \(m\geq -1\) (where \(S=R[It]\), \(t\quad an\) indeterminate). The following conditions are shown to be equivalent: \((i)\quad \omega_ S\) has the expected form; \((ii)\quad \omega_ T\cong \omega_ RT;\) \((iii)\quad \omega_ G\) 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.
Reviewer: W.Bruns

MSC:
13D25 Complexes (MSC2000)
13A15 Ideals and multiplicative ideal theory in commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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