Restricting Homology to Hypersurfaces

  • Luchezar L. Avramov
  • Srikanth B. IyengarEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 242)


This paper concerns the homological properties of a module M over a ring R relative to a presentation \(R\cong P/I\), where P is local ring. It is proved that the Betti sequence of M with respect to P / (f) for a regular element f in I depends only on the class of f in \(I/\mathfrak {n} I\), where \(\mathfrak {n}\) is the maximal ideal of P. Applications to the theory of supports sets in local algebra and in the modular representation theory of elementary abelian groups are presented.


Elementary abelian group Complete intersection Rank variety Support set 

2010 Mathematics Subject Classification

13D07 (primary) 16E45 13D02 13D40 (secondary) 


  1. 1.
    M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, 1969.Google Scholar
  2. 2.
    L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), 71–101.MathSciNetCrossRefGoogle Scholar
  3. 3.
    L. L. Avramov, Infinite free resolutions, Six lectures on commutative algebra (Barcelona, 1996), Progress in Math. 166, Birkhäuser, Basel, 1998; 1–118.Google Scholar
  4. 4.
    L. L. Avramov, R.-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2001), 285–318.MathSciNetCrossRefGoogle Scholar
  5. 5.
    L. L. Avramov, R.-O. Buchweitz, S. B. Iyengar, Class and rank of differential modules, Invent. Math. 169 (2007), 1–35.MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. L. Avramov, H.-B. Foxby, Homological dimensions of unbounded complexes, J. Pure App. Algebra 71 (1991), 129–155.MathSciNetCrossRefGoogle Scholar
  7. 7.
    W. Bruns, J. Herzog, Cohen-Macaulay rings, (Revised ed.), Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1998.Google Scholar
  8. 8.
    J. Carlson, The varieties and the cohomology ring of a module, J. Algebra 85 (1983), 104–143.MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260, (1980), 35–64.MathSciNetCrossRefGoogle Scholar
  10. 10.
    H. B. Foxby, S. Iyengar, Depth and amplitude for unbounded complexes, Commutative Algebra (Grenoble/Lyon, 2001), Contemp. Math. 331, Amer. Math. Soc., Providence, RI, 2003; 119–137.Google Scholar
  11. 11.
    E. M. Friedlander, J. Pevtsova, Representation-theoretic support spaces for finite group schemes, Amer. J. Math. 127 (2005) 379–420.MathSciNetCrossRefGoogle Scholar
  12. 12.
    D. A. Jorgensen, Support sets of pairs of modules, Pacific J. Math. 207 (2002) 393–409.MathSciNetCrossRefGoogle Scholar
  13. 13.
    I. Kaplansky, Fields and rings, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1972.Google Scholar
  14. 14.
    J. Shamash, The Poincaré series of a local ring, J. Algebra 12 (1969), 453–470.MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Suslin, Detection theorem for finite group schemes, J. Pure Appl. Algebra 206 (2006),189–221.MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Suslin, E. Friedlander, C. Bendel, Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), 729–759.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of NebraskaLincolnU.S.A.
  2. 2.Department of MathematicsUniversity of UtahSalt Lake CityU.S.A.

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