Skip to main content

The Chern Numbers and Euler Characteristics of Modules

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.

This is a preview of subscription content, access via your institution.

Notes

  1. 1 The terminology is due to the other five authors.

References

  1. Auslander, M., Buchsbaum, D.: Codimension and multiplicity. Ann. Math. 68, 625–657 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brennan, J., Ulrich, B., Vasconcelos, W.V.: The Buchsbaum–Rim polynomial of a module. J. Algebra 241, 379–392 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  4. Buchsbaum, D., Rim, D.S.: A generalized Koszul complex II. Depth and multiplicity. Trans. Am. Math. Soc. 111, 197–224 (1965)

    Article  MathSciNet  Google Scholar 

  5. Cuong, N.T.: p-standard system of parameters and p-standard ideals in local rings. Acta Math. Vietnam. 20, 145–161 (1995)

    MATH  MathSciNet  Google Scholar 

  6. Cuong, N.T., Schenzel, P., Trung, N.V.: Verallgemeinerte Cohen–Macaulay–Moduln. Math. Nachr. 85, 57–73 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  7. Doering, L.R., Gunston, T., Vasconcelos, W.V.: Cohomological degrees and Hilbert functions of graded modules. Am. J. Math. 120, 493–504 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ghezzi, L., Hong, J., Vasconcelos, W.V.: The signature of the Chern coefficients of local rings. Math. Res. Lett. 16, 279–289 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Goto, S., Hong, J., Vasconcelos, W.V.: The homology of parameter ideals. J. Algebra 368, 271–299 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Goto, S., Nakamura, Y.: Multiplicities and tight closures of parameters. J. Algebra 244, 302–311 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Goto, S., Nishida, K.: Hilbert coefficients and Buchsbaumness of associated graded rings. J. Pure Appl. Algebra 181, 61–74 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Goto, S., Ozeki, K.: Buchsbaumness in local rings possessing constant first Hilbert coefficient of parameters. Nagoya Math. J. 199, 95–105 (2010)

    MATH  MathSciNet  Google Scholar 

  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)

  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)

  16. Hayasaka, F., Hyry, E.: On the Buchsbaum-Rim function of a parameter module. J. Algebra 327, 307–315 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kawasaki, T.: On Cohen–Macaulayfication of certain quasi-projective schemes. J. Math. Soc. Japan 50, 969–991 (1998)

    Article  MathSciNet  Google Scholar 

  18. Linh, C.H.: Upper bound for the Castelnuovo-Mumford regularity of associated graded modules. Commun. Algebra 33, 1817–1831 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mandal, M., Singh, B., Verma, J.K.: On some conjectures about the Chern numbers of filtrations. J. Algebra 325, 147–162 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Matsumura, H.: Commutative Algebra. Benjamin/Cummings, Reading (1980)

  21. Nagata, M.: Local Rings. Interscience, New York (1962)

    MATH  Google Scholar 

  22. Rossi, M.E., Trung, N.V., Valla, G.: Castelnuovo-Mumford regularity and extended degree. Trans. Am. Math. Soc. 355, 1773–1786 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Rossi, M.E., Valla, G.: On the Chern number of a filtration. Rend. Semin. Mat. Univ. Padova 121, 201–222 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  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. Schenzel, P.: Multiplizitäten in verallgemeinerten Cohen–Macaulay–Moduln. Math. Nachr. 88, 295–306 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  26. Serre, J.-P.: Algèbre Locale. Multiplicités. Lecture Notes in Mathematics, vol. 11. Springer, Berlin (1965)

    Google Scholar 

  27. Stückrad, J., Vogel, W.: Toward a theory of Buchsbaum singularities. Am. J. Math. 100, 727–746 (1978)

    Article  MATH  Google Scholar 

  28. Stückrad, J., Vogel, W.: Buchsbaum rings and applications. Springer, Berlin (1986)

    Book  Google Scholar 

  29. Trung, N.V.: Toward a theory of generalized Cohen–Macaulay modules. Nagoya Math. J. 102, 1–49 (1986)

    MathSciNet  Google Scholar 

  30. Vasconcelos, W.V.: The homological degree of a module. Trans. Am. Math. Soc. 350, 1167–1179 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  31. Vasconcelos, W.V.: Integral closure. Springer Monographs in Mathematics, New York (2005)

    MATH  Google Scholar 

  32. Vasconcelos, W.V.: The Chern coefficients of local rings. Michigan Math. J. 57, 725–743 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. Goto.

Additional information

Dedicated to Professors N.V. Trung and G. Valla for their groundbreaking contributions to the theory of Hilbert functions

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ghezzi, L., Goto, S., Hong, J. et al. The Chern Numbers and Euler Characteristics of Modules. Acta Math Vietnam 40, 37–60 (2015). https://doi.org/10.1007/s40306-014-0096-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40306-014-0096-6

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