Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T06:20:51.488Z Has data issue: false hasContentIssue false
  • English
  • Français

The structure of sally modules and Buchsbaumness of associated graded rings

Published online by Cambridge University Press:  11 January 2016

Kazuho Ozeki
Kazuho Ozeki*
Affiliation:
Department of Mathematical Science, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let A be a Noetherian local ring with the maximal ideal m, and let I be an m-primary ideal in A. This paper examines the equality on Hilbert coefficients of I first presented by Elias and Valla, but without assuming that A is a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring of I.

Keywords

13D4013A3013H10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 212 , December 2013 , pp. 97 - 138
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

References

[1] Corso, A., Sally modules of m-primary ideals in local rings, Comm. Algebra 37(2009), 4503–4515. MR 2588863. DOI 10.1080/00927870802266490.Google Scholar
[2] Corso, A., Polini, C., and Vasconcelos, W. V., Multiplicity of the special fiber of blowups, Math. Proc. Cambridge Philos. Soc. 140 (2006), 207–219. MR 2212275. DOI 10.1017/ S0305004105009023.CrossRefGoogle Scholar
[3] Elias, J. and Valla, G., Rigid Hilbert functions, J. Pure Appl. Algebra 71 (1991), 19–41. MR 1107650. DOI 10.1016/0022-4049(91)90038–4.Google Scholar
[4] Ghezzi, L., Goto, S., Hong, J., Ozeki, K., Phuong, T. T., and Vasconcelos, W. V., Variation of the first Hilbert coefficients of parameters with a common integral closure, J. Pure Appl. Algebra 216 (2012), 216–232. MR 2826435. DOI 10.1016/j.jpaa.2011. 06.008.Google Scholar
[5] Goto, S., Noetherian local rings with Buchsbaum associated graded rings, J. Algebra 86 (1984), 336–384. MR 0732255. DOI 10.1016/0021-8693(84)90037–1.CrossRefGoogle Scholar
[6] Goto, S. and Nishida, K., Hilbert coefficients and Buchsbaumness of associated graded rings, J. Pure Appl. Algebra 181 (2003), 61–74. MR 1971805. DOI 10.1016/S0022-4049(02)00325-0.CrossRefGoogle Scholar
[7] Goto, S., Nishida, K., and Ozeki, K., Sally modules of rank one, Michigan Math. J. 57 (2008), 359–381. MR 2492458. DOI 10.1307/mmj/1220879414.Google Scholar
[8] Goto, S. and Ozeki, K., The structure of Sally modules—towards a theory of non-Cohen–Macaulay cases, J. Algebra 324 (2010), 2129–2165. MR 2684135. DOI 10. 1016/j.jalgebra.2010.07.017.Google Scholar
[9] Goto, S. and Ozeki, K., “Uniform bounds for Hilbert coefficients of parameters” in Commutative Algebra and Its Connections to Geometry (Olinda, 2009), Contemp. Math. 555, Amer. Math. Soc., Providence, 2011, 97–118. MR 2882677. DOI 10.1090/conm/555/ 10992.Google Scholar
[10] Goto, S. and Shimoda, Y., On Rees algebras over Buchsbaum rings, J. Math. Kyoto Univ. 20 (1980), 691–708. MR 0592354.Google Scholar
[11] Guerrieri, A. and Rossi, M. E., Hilbert coefficients of Hilbert filtrations, J. Algebra 199 (1998), 40–61. MR 1489353. DOI 10.1006/jabr.1997.7194.CrossRefGoogle Scholar
[12] Huckaba, S. and Marley, T., Depth formulas for certain graded rings associated to an ideal, Nagoya Math. J. 133 (1994), 57–69. MR 1266362.CrossRefGoogle Scholar
[13] Huneke, C., On the symmetric and Rees algebra of an ideal generated by a d-sequence, J. Algebra 62 (1980), 268–275. MR 0563225. DOI 10.1016/0021-8693 (80)90179-9.Google Scholar
[14] Huneke, C., Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), 293–318. MR 0894879. DOI 10.1307/mmj/1029003560.Google Scholar
[15] Nagata, M., A proof of the theorem of Eakin-Nagata, Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), 238–239. MR 1137916.Google Scholar
[16] Ozeki, K., The equality of Elias–Valla and the associated graded ring of maximal ideals, J. Pure Appl. Algebra 216 (2012), 1306–1317. MR 2890504. DOI 10.1016/j. jpaa.2012.01.004.Google Scholar
[17] Ratliff, L. J. and Rush, D. E., Two notes on reductions of ideals, Indiana Univ. Math. J. 27 (1978), 929–934. MR 0506202.Google Scholar
[18] Rossi, M. E. and Valla, G., Hilbert Functions of Filtered Modules, Lect. Notes Unione Mat. Ital. 9, Springer, Berlin, 2010. MR 2723038. DOI 10.1007/978-3-642-14240-6.Google Scholar
[19] Schenzel, P., Multiplizitäten in verallgemeinerten Cohen–Macaulay-Moduln, Math. Nachr. 88 (1979), 295–306. MR 0543409. DOI 10.1002/mana.19790880122.Google Scholar
[20] Stückrad, J. and Vogel, W., Buchsbaum Rings and Applications: An Interaction between Algebra, Geometry and Topology, Springer, Berlin, 1986. MR 0881220.Google Scholar
[21] Trung, N. V., Toward a theory of generalized Cohen–Macaulay modules, Nagoya Math. J. 102 (1986), 1–49. MR 0846128.Google Scholar
[22] Trung, N. V., Reduction exponent and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc. 101 (1987), 229–236. MR 0902533. DOI 10.2307/2045987.Google Scholar
[23] Vasconcelos, W. V., “Hilbert functions, analytic spread, and Koszul homology” in Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra (South Hadley, Mass., 1992), Contemp. Math. 159, Amer. Math. Soc., Providence, 1994, 401–422. MR 1266195. DOI 10.1090/conm/159/01520.Google Scholar