# zbMATH — the first resource for mathematics

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.)
Full Text:
##### References:
  Buchsbaum, D; Eisenbud, D, What makes a complex exact?, J. algebra, 25, 259-268, (1973) · Zbl 0264.13007  Buchsbaum, D; Eisenbud, D, Algebra structure for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. math., 99, 447-485, (1977) · Zbl 0373.13006  Eisenbud, D; Huneke, C, Cohen-Macaulay Rees algebras and their specializations, J. algebra, 81, 202-224, (1983) · Zbl 0528.13024  Goto, S, The divisor class group of a certain krulldomain, J. math. Kyoto univ., 17, 47-50, (1977) · Zbl 0368.13015  Goto, S; Shimoda, Y, On the Rees algebras of Cohen-Macaulay local rings, commutative algebra (analytic methods), (), 201-231  Grothe, U; Herrmann, M; Orbanz, U, Graded Cohen-Macaulay rings associated to equimultiple ideals, Math. Z., 186, 531-556, (1984) · Zbl 0541.13010  \scJ. Herzog, Komplexe, Auflösungen und Dualität in der lokalen Algebra, Habilitationsschrift, Universität Regensburg.  Herzog, J; Kunz, E, Der kanonische modul eines Cohen-Macaulay rings, () · Zbl 0231.13009  Herzog, J; Simis, A; Vasconcelos, W.V, Approximation complexes of blowing-up rings, I, J. algebra, 74, 466-493, (1982) · Zbl 0484.13006  Herzog, J; Simis, A; Vasconcelos, W.V, Approximation complexes of blowing-up rings, II, J. algebra, 82, 53-83, (1983) · Zbl 0515.13018  Herzog, J; Simis, A; Vasconcelos, W.V, (), 79-169  Herzog, J; Simis, A; Vasconcelos, W.V, On the arithmetic and homology of algebras of linear type, Trans. amer. math. soc., 283, 661-683, (1984) · Zbl 0541.13005  Herzog, J; Vasconcelos, W.V; Villarreal, R, Ideals with sliding depth, Nagoya math. J., 99, 159-172, (1985) · Zbl 0561.13014  Herzog, J; Vasconcelos, W.V, On the divisor class group of Rees-algebras, J. algebra, 93, 182-188, (1985) · Zbl 0562.13003  Hochster, M, Criteria for equality of ordinary and symbolic powers of an ideal, Math. Z., 133, 53-65, (1973) · Zbl 0251.13012  Huneke, C, The theory of d-sequences and powers of ideals, Adv. in math., 46, 3, 249-279, (1982) · Zbl 0505.13004  Huneke, C, On the associated graded ring of an ideal, Illinois J. math., 26, 1, 121-137, (1982) · Zbl 0479.13008  Huneke, C, Strongly Cohen-Macaulay schemes and residual intersections, Trans. amer. math. soc., 277, 739-763, (1983) · Zbl 0514.13011  Huneke, C, Linkage and the Koszul homology of ideals, Amer. J. math., 104, 1043-1062, (1982) · Zbl 0505.13003  Huneke, C, On the symmetric and Rees-algebras of an ideal generated by a d-sequence, J. algebra, 62, 268-275, (1980) · Zbl 0439.13001  Peskine, C; Szpiro, L, Liaison des varietés algébriques, Invent. math., 26, 271-302, (1973) · Zbl 0298.14022  Sally, J, On the associated graded ring of a local Cohen-Macaulay ring, J. math. Kyoto univ., 17, 19-21, (1977) · Zbl 0353.13017  Sally, J, Cohen-Macaulay local rings of maximal embedding dimension, J. algebra, 56, 168-183, (1979) · Zbl 0401.13016  Sally, J, Numbers of generators of ideals in local rings, () · Zbl 0395.13010  Sanders, H, Cohen-Macaulay properties of the Koszul homology, Manuscripta math., 55, 343-357, (1986) · Zbl 0644.13005  Simis, A; Vasconcelos, W.V, The syzygies of the conormal module, Amer. J. math., 103, 203-224, (1981) · Zbl 0467.13009  Steurich, M, On rings with linear resolutions, J. algebra, 75, 1, 178-197, (1982) · Zbl 0491.13015  Valla, G, On the symmetric and Rees-algebras of an ideal, Manuscripta math., 30, 239, (1980) · Zbl 0439.13002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.