Advertisement

Acta Mathematica Vietnamica

, Volume 40, Issue 1, pp 37–60 | Cite as

The Chern Numbers and Euler Characteristics of Modules

  • L. Ghezzi
  • S. Goto
  • J. Hong
  • K. Ozeki
  • T. T. Phuong
  • W. V. Vasconcelos
Article

Abstract

The set of the first Hilbert coefficients of parameter ideals relative to a module—its Chern coefficients—over a local Noetherian ring codes for considerable information about its structure–noteworthy properties such as that of Cohen-Macaulayness, Buchsbaumness, and of having finitely generated local cohomology. The authors have previously studied the ring case. By developing a robust setting to treat these coefficients for unmixed rings and modules, the case of modules is analyzed in a more transparent manner. Another series of integers arise from partial Euler characteristics and are shown to carry similar properties of the module. The technology of homological degree theory is also introduced in order to derive bounds for these two sets of numbers.

Keywords

Hilbert function Hilbert coefficient Parameter ideal Local cohomology Euler characteristic Cohen-Macaulay module Vasconcelos module Generalized Cohen-Macaulay module Buchsbaum module 

Mathematics Subject Classification (2010)

13H10 13H15 13A30 

Notes

Acknowledgments

The first author is partially supported by a grant from the City University of New York PSC-CUNY Research Award Program-44. The second author is partially supported by Grant-in-Aid for Scientific Researches (C) in Japan (19540054). The fourth author is supported by a grant from MIMS (Meiji Institute for Advanced Study of Mathematical Sciences). The fifth author is supported by JSPS Ronpaku (Dissertation of PhD) Program. The last author is partially supported by the NSF

References

  1. 1.
    Auslander, M., Buchsbaum, D.: Codimension and multiplicity. Ann. Math. 68, 625–657 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Brennan, J., Ulrich, B., Vasconcelos, W.V.: The Buchsbaum–Rim polynomial of a module. J. Algebra 241, 379–392 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge University Press, Cambridge (1993)Google Scholar
  4. 4.
    Buchsbaum, D., Rim, D.S.: A generalized Koszul complex II. Depth and multiplicity. Trans. Am. Math. Soc. 111, 197–224 (1965)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cuong, N.T.: p-standard system of parameters and p-standard ideals in local rings. Acta Math. Vietnam. 20, 145–161 (1995)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Cuong, N.T., Schenzel, P., Trung, N.V.: Verallgemeinerte Cohen–Macaulay–Moduln. Math. Nachr. 85, 57–73 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Doering, L.R., Gunston, T., Vasconcelos, W.V.: Cohomological degrees and Hilbert functions of graded modules. Am. J. Math. 120, 493–504 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Ghezzi, L., Goto, S., Hong, J., Ozeki, K., Phuong, T.T., Vasconcelos, W.V.: Cohen–Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals. J. Lond. Math. Soc. 81, 679–695 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Ghezzi, L., Hong, J., Vasconcelos, W.V.: The signature of the Chern coefficients of local rings. Math. Res. Lett. 16, 279–289 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Goto, S., Hong, J., Vasconcelos, W.V.: The homology of parameter ideals. J. Algebra 368, 271–299 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Goto, S., Nakamura, Y.: Multiplicities and tight closures of parameters. J. Algebra 244, 302–311 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Goto, S., Nishida, K.: Hilbert coefficients and Buchsbaumness of associated graded rings. J. Pure Appl. Algebra 181, 61–74 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Goto, S., Ozeki, K.: Buchsbaumness in local rings possessing constant first Hilbert coefficient of parameters. Nagoya Math. J. 199, 95–105 (2010)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Goto, S., Ozeki, K.: Uniform bounds for Hilbert coefficients of parameters. In: Commutative Algebra and its Connections to Geometry, 97–118, Contemp. Math, 555, Am. Math. Soc., Providence, RI (2011)Google Scholar
  15. 15.
    Gulliksen, T., Levin, G.: Homology of local rings, Queen’s Paper in Pure and Applied Mathematics, No. 20, Queen’s University, Kingston, Ont (1969)Google Scholar
  16. 16.
    Hayasaka, F., Hyry, E.: On the Buchsbaum-Rim function of a parameter module. J. Algebra 327, 307–315 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kawasaki, T.: On Cohen–Macaulayfication of certain quasi-projective schemes. J. Math. Soc. Japan 50, 969–991 (1998)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Linh, C.H.: Upper bound for the Castelnuovo-Mumford regularity of associated graded modules. Commun. Algebra 33, 1817–1831 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Mandal, M., Singh, B., Verma, J.K.: On some conjectures about the Chern numbers of filtrations. J. Algebra 325, 147–162 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Matsumura, H.: Commutative Algebra. Benjamin/Cummings, Reading (1980)Google Scholar
  21. 21.
    Nagata, M.: Local Rings. Interscience, New York (1962)zbMATHGoogle Scholar
  22. 22.
    Rossi, M.E., Trung, N.V., Valla, G.: Castelnuovo-Mumford regularity and extended degree. Trans. Am. Math. Soc. 355, 1773–1786 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Rossi, M.E., Valla, G.: On the Chern number of a filtration. Rend. Semin. Mat. Univ. Padova 121, 201–222 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Rossi, M.E., Valla, G.: Hilbert functions of filtered modules. Lecture Notes of the Unione Matematica Italiana, vol. 9. Springer, Berlin (2010)Google Scholar
  25. 25.
    Schenzel, P.: Multiplizitäten in verallgemeinerten Cohen–Macaulay–Moduln. Math. Nachr. 88, 295–306 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Serre, J.-P.: Algèbre Locale. Multiplicités. Lecture Notes in Mathematics, vol. 11. Springer, Berlin (1965)Google Scholar
  27. 27.
    Stückrad, J., Vogel, W.: Toward a theory of Buchsbaum singularities. Am. J. Math. 100, 727–746 (1978)CrossRefzbMATHGoogle Scholar
  28. 28.
    Stückrad, J., Vogel, W.: Buchsbaum rings and applications. Springer, Berlin (1986)CrossRefGoogle Scholar
  29. 29.
    Trung, N.V.: Toward a theory of generalized Cohen–Macaulay modules. Nagoya Math. J. 102, 1–49 (1986)MathSciNetGoogle Scholar
  30. 30.
    Vasconcelos, W.V.: The homological degree of a module. Trans. Am. Math. Soc. 350, 1167–1179 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Vasconcelos, W.V.: Integral closure. Springer Monographs in Mathematics, New York (2005)zbMATHGoogle Scholar
  32. 32.
    Vasconcelos, W.V.: The Chern coefficients of local rings. Michigan Math. J. 57, 725–743 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014

Authors and Affiliations

  • L. Ghezzi
    • 1
  • S. Goto
    • 2
  • J. Hong
    • 3
  • K. Ozeki
    • 2
  • T. T. Phuong
    • 4
  • W. V. Vasconcelos
    • 5
  1. 1.Department of MathematicsNew York City College of Technology-CunyBrooklynUSA
  2. 2.Department of Mathematics, School of Science and TechnologyMeiji UniversityKawasakiJapan
  3. 3.Department of MathematicsSouthern Connecticut State UniversityNew HavenUSA
  4. 4.Department of Information Technology and Applied MathematicsTon Duc Thang UniversityHo Chi Minh CityVietnam
  5. 5.Department of MathematicsRutgers UniversityPiscatawayUSA

Personalised recommendations