Good ideals in Gorenstein local rings
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- by Shiro Goto, Sin-Ichiro Iai and Kei-ichi Watanabe PDF
- Trans. Amer. Math. Soc. 353 (2001), 2309-2346 Request permission
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.References
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Jean Giraud, Improvement of Grauert-Riemenschneider’s theorem for a normal surface, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 4, 13–23 (1983) (English, with French summary). MR 694126, DOI 10.5802/aif.892
- Shiro Goto and Sam Huckaba, On graded rings associated to analytic deviation one ideals, Amer. J. Math. 116 (1994), no. 4, 905–919. MR 1287943, DOI 10.2307/2375005
- Shiro Goto and Koji Nishida, The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations, American Mathematical Society, Providence, RI, 1994. Mem. Amer. Math. Soc. 110 (1994), no. 526. MR 1287443
- Shiro Goto and Yasuhiro Shimoda, On the Rees algebras of Cohen-Macaulay local rings, Commutative algebra (Fairfax, Va., 1979) Lecture Notes in Pure and Appl. Math., vol. 68, Dekker, New York, 1982, pp. 201–231. MR 655805
- Shiro Goto and Keiichi Watanabe, On graded rings. I, J. Math. Soc. Japan 30 (1978), no. 2, 179–213. MR 494707, DOI 10.2969/jmsj/03020179
- Craig Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), no. 2, 293–318. MR 894879, DOI 10.1307/mmj/1029003560
- M. Hochster and John A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020–1058. MR 302643, DOI 10.2307/2373744
- M. Herrmann, C. Huneke, and J. Ribbe, On reduction exponents of ideals with Gorenstein formring, Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 3, 449–463. MR 1357642, DOI 10.1017/S0013091500019258
- M. Herrmann, S. Ikeda, and U. Orbanz, Equimultiplicity and blowing up, Springer-Verlag, Berlin, 1988. An algebraic study; With an appendix by B. Moonen. MR 954831, DOI 10.1007/978-3-642-61349-4
- Jürgen Herzog and Ernst Kunz (eds.), Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Mathematics, Vol. 238, Springer-Verlag, Berlin-New York, 1971. Seminar über die lokale Kohomologietheorie von Grothendieck, Universität Regensburg, Wintersemester 1970/1971. MR 0412177
- Craig Huneke and Irena Swanson, Cores of ideals in $2$-dimensional regular local rings, Michigan Math. J. 42 (1995), no. 1, 193–208. MR 1322199, DOI 10.1307/mmj/1029005163
- Shin Ikeda, On the Gorensteinness of Rees algebras over local rings, Nagoya Math. J. 102 (1986), 135–154. MR 846135, DOI 10.1017/S0027763000000489
- Masahide Kato, Riemann-Roch theorem for strongly pseudoconvex manifolds of dimension $2$, Math. Ann. 222 (1976), no. 3, 243–250. MR 412468, DOI 10.1007/BF01362581
- Henry B. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977), no. 6, 1257–1295. MR 568898, DOI 10.2307/2374025
- Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239, DOI 10.1007/BF02684604
- Joseph Lipman, Stable ideals and Arf rings, Amer. J. Math. 93 (1971), 649–685. MR 282969, DOI 10.2307/2373463
- Joseph Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), no. 1, 151–207. MR 491722, DOI 10.2307/1971141
- Joseph Lipman, Adjoints of ideals in regular local rings, Math. Res. Lett. 1 (1994), no. 6, 739–755. With an appendix by Steven Dale Cutkosky. MR 1306018, DOI 10.4310/MRL.1994.v1.n6.a10
- Joseph Lipman and Bernard Teissier, Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), no. 1, 97–116. MR 600418
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- Idun Reiten, The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc. 32 (1972), 417–420. MR 296067, DOI 10.1090/S0002-9939-1972-0296067-7
- Judith D. Sally, On the associated graded ring of a local Cohen-Macaulay ring, J. Math. Kyoto Univ. 17 (1977), no. 1, 19–21. MR 450259, DOI 10.1215/kjm/1250522807
- Judith D. Sally, Numbers of generators of ideals in local rings, Marcel Dekker, Inc., New York-Basel, 1978. MR 0485852
- Kishor Shah, On the Cohen-Macaulayness of the fiber cone of an ideal, J. Algebra 143 (1991), no. 1, 156–172. MR 1128652, DOI 10.1016/0021-8693(91)90257-9
- Siegfried Bosch, Multiplikative Untergruppen in abeloiden Mannigfaltigkeiten, Math. Ann. 239 (1979), no. 2, 165–183 (German). MR 519012, DOI 10.1007/BF01420374
Additional Information
- Shiro Goto
- Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 214-8571 Japan
- MR Author ID: 192104
- 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
- MR Author ID: 216208
- Email: watanabe@math.chs.nihon.-u.ac.jp
- 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)
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1814072