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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Good ideals in Gorenstein local rings


Authors: Shiro Goto, Sin-Ichiro Iai and Kei-ichi Watanabe
Journal: Trans. Amer. Math. Soc. 353 (2001), 2309-2346
MSC (2000): Primary 13A30; Secondary 13H10
DOI: https://doi.org/10.1090/S0002-9947-00-02694-5
Published electronically: November 29, 2000
MathSciNet review: 1814072
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Abstract:

Let $I$ be an $\mathfrak{m}$-primary ideal in a Gorenstein local ring ($A$, $\mathfrak{m}$) with $\dim A = d$, and assume that $I$ contains a parameter ideal $Q$ in $A$ as a reduction. We say that $I$ is a good ideal in $A$ if $G = \sum _{n \geq 0} I^{n}/I^{n+1}$ is a Gorenstein ring with $\mathrm{a} (G) = 1 - d$. The associated graded ring $G$ of $I$ is a Gorenstein ring with $\mathrm{a}(G) = -d$ if and only if $I = Q$. Hence good ideals in our sense are good ones next to the parameter ideals $Q$ in $A$. A basic theory of good ideals is developed in this paper. We have that $I$ is a good ideal in $A$ if and only if $I^{2} = QI$ and $I = Q : I$. First a criterion for finite-dimensional Gorenstein graded algebras $A$ over fields $k$ to have nonempty sets $\mathcal{X}_{A}$ of good ideals will be given. Second in the case where $d = 1$ we will give a correspondence theorem between the set $\mathcal{X}_{A}$ and the set $\mathcal{Y}_{A}$ of certain overrings of $A$. A characterization of good ideals in the case where $d = 2$ will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set $\mathcal{X}_{A}$ of good ideals in $A$ heavily depends on $d = \dim A$. The set $\mathcal{X}_{A}$ may be empty if $d \leq 2$, while $\mathcal{X}_{A}$ is necessarily infinite if $d \geq 3$ and $A$contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring $k[X_{1},X_{2},X_{3}]$ in three variables over a field $k$. Examples are given to illustrate the theorems.


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  • [BH] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge studies in advanced mathematics, vol. 39, Cambridge University Press, 1993. MR 95h:13020
  • [G] J. Giraud, Improvement of Grauert-Riemenschneider's Theorem for a normal surface, Ann. Inst. Fourier 32 (1982), fasc. 4, 13-23. MR 84f:14025
  • [GH] S. Goto and S. Huckaba, On graded rings associated to analytic deviation one ideals, Amer. J. Math. 116 (1994), 905-919. MR 95h:13003
  • [GN] S. Goto and K. Nishida, The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations, Mem. Amer. Math. Soc. 110 (1994), no. 526. MR 95b:13001
  • [GS] S. Goto and Y. Shimoda, On the Rees algebras of Cohen-Macaulay local rings, Commutative Algebra, Analytic Methods (R. N. Draper, ed.), Lecture Notes in Pure and Applied Mathematics, vol. 68, Marcel Dekker, Inc., New York and Basel, 1982, pp. 201-231. MR 84a:13021
  • [GW] S. Goto and K. Watanabe, On graded rings I, J. Math. Soc. Japan 30 (1978), 179-213. MR 81m:13021
  • [H] C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), 293-318. MR 89b:13037
  • [HE] M. Hochster and J. A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of of determinantal loci, Amer. J. Math. 93 (1971), 1020-1058. MR 46:1787
  • [HHR] M. Herrmann, C. Huneke, and J. Ribbe, On reduction exponents of ideals with Gorenstein form rings, Proc. Edinburgh Math. Soc. 38 (1995), 449-463. MR 96i:13007
  • [HIO] M. Herrmann, S. Ikeda, and U. Orbanz, Equimultiplicity and blowing up, Springer-Verlag, 1988.MR 89g:13012
  • [HK] J. Herzog and E. Kunz (eds.), Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Mathematics, vol. 238, Springer-Verlag, 1971. MR 54:304
  • [HS] C. Huneke and I. Swanson, Cores of ideals in 2-dimensional regular local rings, Michigan Math. J. 42 (1995), 193-208. MR 96j:13021
  • [I] S. Ikeda, On the Gorensteinness of Rees algebras over local rings, Nagoya Math. J 102 (1986), 135-154. MR 87j:13031
  • [K] Ma. Kato, Riemann-Roch theorem for strongly pseudoconvex manifolds of dimension 2, Math. Ann. 222 (1976), 243-250. MR 54:594
  • [La] H. B. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977), 1257-1295. MR 58:27961
  • [L1] J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math. IHES 36 (1969), 195-279. MR 43:1986
  • [L2] J. Lipman, Stable ideals and Arf rings, Amer. J. Math. 93 (1971), 649-685. MR 44:203
  • [L3] J. Lipman, Desingularization of two-dimensional schemes, Ann. of Math. 107 (1978), 151-207. MR 58:10924
  • [L4] J. Lipman, Adjoints of ideals in regular local rings, Mathematical Research Letters 1 (1994), 739-755. MR 95k:13028
  • [LT] J. Lipman and B. Teissier, Pseudo-rational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), 97-116. MR 82f:14004
  • [NR] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Math. Proc. Cambridge Philos. Soc. 50 (1954), 145-158. MR 15:596a
  • [R] I. Reiten, The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc. 32 (1972), 417-420. MR 45:5128
  • [S1] J. D. Sally, On the associated graded ring of a local Cohen-Macaulay ring, J. Math. Kyoto Univ. 17 (1977), 19-21. MR 56:8555
  • [S2] J. Sally, Numbers of generators of ideals in local rings, Lecture Notes in Pure and Applied Mathematics, vol. 35, Marcel Dekker, Inc., New York and Basel, 1978. MR 58:5654
  • [Sh] K. Shah, On the Cohen-Macaulayness of the fiber cone of an ideal, J. Alg. 143 (1991), 156-172. MR 92k:13014
  • [VV] P. Valabrega and G. Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93-101. MR 80d:14014

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Additional Information

Shiro Goto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 214-8571 Japan
Email: goto@math.meiji.ac.jp

Sin-Ichiro Iai
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 214-8571 Japan
Email: s-iai@math.meiji.ac.jp

Kei-ichi Watanabe
Affiliation: Department of Mathematics, Nihon University, 156-8550 Japan
Email: watanabe@math.chs.nihon.-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-00-02694-5
Keywords: Rees algebra, associated graded ring, Cohen-Macaulay ring, Gorenstein ring, $\mathrm{a}$-invariant
Received by editor(s): July 25, 1999
Published electronically: November 29, 2000
Additional Notes: The first and third authors are supported by the Grant-in-Aid for Scientific Researches in Japan (C(2), No. 11640049 and 10640042, respectively)
Article copyright: © Copyright 2000 American Mathematical Society

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