Koszul cycles and Golod rings

  • Jürgen Herzog
  • Rasoul Ahangari Maleki


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 


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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).


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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

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