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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

13D25 Complexes (MSC2000)
13A15 Ideals and multiplicative ideal theory in commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Full Text: DOI
[1] Buchsbaum, D; Eisenbud, D, What makes a complex exact?, J. algebra, 25, 259-268, (1973) · Zbl 0264.13007
[2] 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
[3] Eisenbud, D; Huneke, C, Cohen-Macaulay Rees algebras and their specializations, J. algebra, 81, 202-224, (1983) · Zbl 0528.13024
[4] Goto, S, The divisor class group of a certain krulldomain, J. math. Kyoto univ., 17, 47-50, (1977) · Zbl 0368.13015
[5] Goto, S; Shimoda, Y, On the Rees algebras of Cohen-Macaulay local rings, commutative algebra (analytic methods), (), 201-231
[6] Grothe, U; Herrmann, M; Orbanz, U, Graded Cohen-Macaulay rings associated to equimultiple ideals, Math. Z., 186, 531-556, (1984) · Zbl 0541.13010
[7] \scJ. Herzog, Komplexe, Auflösungen und Dualität in der lokalen Algebra, Habilitationsschrift, Universität Regensburg.
[8] Herzog, J; Kunz, E, Der kanonische modul eines Cohen-Macaulay rings, () · Zbl 0231.13009
[9] Herzog, J; Simis, A; Vasconcelos, W.V, Approximation complexes of blowing-up rings, I, J. algebra, 74, 466-493, (1982) · Zbl 0484.13006
[10] Herzog, J; Simis, A; Vasconcelos, W.V, Approximation complexes of blowing-up rings, II, J. algebra, 82, 53-83, (1983) · Zbl 0515.13018
[11] Herzog, J; Simis, A; Vasconcelos, W.V, (), 79-169
[12] 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
[13] Herzog, J; Vasconcelos, W.V; Villarreal, R, Ideals with sliding depth, Nagoya math. J., 99, 159-172, (1985) · Zbl 0561.13014
[14] Herzog, J; Vasconcelos, W.V, On the divisor class group of Rees-algebras, J. algebra, 93, 182-188, (1985) · Zbl 0562.13003
[15] Hochster, M, Criteria for equality of ordinary and symbolic powers of an ideal, Math. Z., 133, 53-65, (1973) · Zbl 0251.13012
[16] Huneke, C, The theory of d-sequences and powers of ideals, Adv. in math., 46, 3, 249-279, (1982) · Zbl 0505.13004
[17] Huneke, C, On the associated graded ring of an ideal, Illinois J. math., 26, 1, 121-137, (1982) · Zbl 0479.13008
[18] Huneke, C, Strongly Cohen-Macaulay schemes and residual intersections, Trans. amer. math. soc., 277, 739-763, (1983) · Zbl 0514.13011
[19] Huneke, C, Linkage and the Koszul homology of ideals, Amer. J. math., 104, 1043-1062, (1982) · Zbl 0505.13003
[20] 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
[21] Peskine, C; Szpiro, L, Liaison des varietés algébriques, Invent. math., 26, 271-302, (1973) · Zbl 0298.14022
[22] Sally, J, On the associated graded ring of a local Cohen-Macaulay ring, J. math. Kyoto univ., 17, 19-21, (1977) · Zbl 0353.13017
[23] Sally, J, Cohen-Macaulay local rings of maximal embedding dimension, J. algebra, 56, 168-183, (1979) · Zbl 0401.13016
[24] Sally, J, Numbers of generators of ideals in local rings, () · Zbl 0395.13010
[25] Sanders, H, Cohen-Macaulay properties of the Koszul homology, Manuscripta math., 55, 343-357, (1986) · Zbl 0644.13005
[26] Simis, A; Vasconcelos, W.V, The syzygies of the conormal module, Amer. J. math., 103, 203-224, (1981) · Zbl 0467.13009
[27] Steurich, M, On rings with linear resolutions, J. algebra, 75, 1, 178-197, (1982) · Zbl 0491.13015
[28] Valla, G, On the symmetric and Rees-algebras of an ideal, Manuscripta math., 30, 239, (1980) · Zbl 0439.13002
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