Koszul cycles and Golod rings

Article
  • 37 Downloads

Abstract

Let S be the power series ring or the polynomial ring over a field K in the variables \(x_1,\ldots ,x_n\), and let \(R=S/I\), where I is proper ideal which we assume to be graded if S is the polynomial ring. We give an explicit description of the cycles of the Koszul complex whose homology classes generate the Koszul homology of \(R=S/I\) with respect to \(x_1,\ldots ,x_n\). The description is given in terms of the data of the free S-resolution of R. The result is used to determine classes of Golod ideals, among them proper ordinary powers and proper symbolic powers of monomial ideals. Our theory is also applied to stretched local rings.

Mathematics Subject Classification

13A02 13D40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The second author was jointly supported by the Iran National Science Foundation (INSF) and Alzahra University grant No. 95001343. This research was also in part supported by a grant from IPM (No. 95130111).

References

  1. 1.
    Anick, D.J.: A counterexample to a conjecture of Serre. Ann. Math. 115(1), 1–33 (1982)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aramova, A., Herzog, J.: Koszul cycles and Eliahou–Kervaire type resolutions. J. Algebra 181, 347–370 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Conca, A., Rossi, M.E., Valla, G.: Gröbner flags and Gorenstein algebras. Compos. Math. 129(1), 95–121 (2001)CrossRefMATHGoogle Scholar
  4. 4.
    Croll, A., Dellaca, R., Gupta, A., Hoffmeier, J., Mukundan, V., Tracy, Denise R., Sega, Liana M., Sosa, G., Thompson, P.: Detecting Koszulness and related homological properties from the algebra structure of Koszul homology. arXiv:1611.04001 [math.AC]
  5. 5.
    Fröberg, R.: Connections between a local ring and its associated graded ring. J. Algebra 111, 300–305 (1987)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fröberg, R.: Koszul algebras. In: Dobbs, D.E., Fontana, M., Kabbaij, S.-E. (eds.) Advanced in Commutative Ring Theory (Fez 1997) 337–350, Lecture Notes in Pure and Application Mathematics (1999)Google Scholar
  7. 7.
    Herzog, J.: Canonical Koszul cycles. Notas de Investigación 6, 33–41 (1992)MathSciNetMATHGoogle Scholar
  8. 8.
    Herzog, J., Huneke, C.: Ordinary and symbolic powers are Golod. Adv. Math. 246, 89–99 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Maleki, R.A.: Golod property of powers of ideals and of ideals with linear resolutions. arXiv:1510.04435 [math.AC]
  10. 10.
    Sally, J.D.: The Poincare series of stretched Cohen–Macaulay rings. Can. J. Math. 32, 1261–1265 (1980)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Serre, J.P.: Algèbre locale. Multiplicités, Lecture Notes Mathematics, vol. 11. Springer, Berlin (1965)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität Duisburg-EssenEssenGermany
  2. 2.School of Mathematics, Institute for Research in Fundamental Sciences (IPM)TehranIran
  3. 3.Department of MathematicsAlzahra UniversityTehranIran

Personalised recommendations