Hilbert series of finitely presented algebras

  • David Anick
  • Clas Löfwall
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1183)


Let l denote the collection of all Hilbert series of finitely presented connected graded algebras over a field k. What can we say about the set l? This paper addresses itself to that question. In 1974 Govorov [Go-2] conjectured that only rational power series belonged to l. This conjecture was first disproved by Shearer [Sh], using methods which we will generalize and extend in this paper. We will also show that the set l is countable and derive some of its properties.


Local Ring Algebra Structure Hilbert Series Homogeneous Element Grade Vector Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [An-1]
    ANICK, D., A counterexample to a conjecture of Serre, Ann. Math. 115, 1982 1–33. Correction: Ann. Math., 116, 1983, 661.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [An-2]
    ANICK, D., The smallest singularity of a Hilbert series, Math. Scand., 51, 1982, 35–44.MathSciNetzbMATHGoogle Scholar
  3. [Be]
    BERGMAN, G.M., The diamond lemma for ring theory, Advances in Math., 29, 1978, 178–218.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [Go-1]
    GOVOROV, V.E., Graded algebras, Math. Notes of the Acad. Sc. of the USSR, 12, 1972, 552–556.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Go-2]
    GOVOROV, V.E., On the dimension of graded algebras, Math. Notes of the Acad. Sc. of the USSR, 14, 1973, 678–682.CrossRefzbMATHGoogle Scholar
  6. [Ja]
    JACOBSSON, C., On the double Poincaré series of the enveloping algebras of certain graded Lie algebras, Math. Scand. 51, 1982, 45–58.MathSciNetzbMATHGoogle Scholar
  7. [Le]
    LEMAIRE, J.-M., Algèbres connexes et homologie des espaces de lacets, Lecture Notes in Mathematics, 422, 1974, Springer-Verlag, Berlin, Heidelberg, New York.zbMATHGoogle Scholar
  8. [Lö]
    LÖFWALL, C., On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra, these proceedings.Google Scholar
  9. [Ro]
    ROOS, J.-E., Relations between the Poincaré-Betti series of loop spaces and of local rings, Lecture Notes in Mathematics, 740, 1979, 285–322, Springer-Verlag, Berlin, Heidelberg, New York.Google Scholar
  10. [Sh]
    SHEARER, J.B., A graded algebra with non-rational Hilbert series, Journ. of Algebra, 62, 1980, 228–231.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • David Anick
    • 1
  • Clas Löfwall
    • 2
  1. 1.Department of MathematicsMass. Institute of TechnologyCambridgeUSA
  2. 2.Department of MathematicsUniversity of StockholmStockholmSweden

Personalised recommendations