Gradings of Polynomial Rings

  • Peter Russell

Abstract

These brief notes have their origin in a conversation I had with Abhyankar during a pleasant walk along the Wabash a few years ago. While I tried to explain some ideas on torus actions on affine spaces Abhyankar, in that inimitable way he has to concentrate one’s mind, feigned innocence as to such fancy notions, but was quite happy to listen when I proposed talking about gradings of polynomial rings instead. The two topics cover exactly the same ground from two different points of view, equally valid and valuable. Torus actions also provide an uncomplicated introduction to the larger subject of reductive group actions. I gave my conference talk hoping that such an introduction in the readily understood language of gradings would be appreciated by some of the non-experts in the rather diverse audience Abhyankar’s birthday was bound to bring together. This is a slightly expanded version of my talk. It is entirely expository in nature and not meant for the specialists.

Keywords

Manifold 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Abhyankar, S. S. and Moh, T. T., Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166.MathSciNetMATHGoogle Scholar
  2. [2]
    Bass, H. and Haboush, W., Linearizing certain reductive group actions, Trans. Am. Math. Soc. Vol.292, No.2, (1985), 463–481.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Bialynicki-Birula, A., Remarks on the action of an algebraic torus on kn, I and II, Bull. Acad. Pol. Sci., 14 (1966), 177–181 and 15 (1967), 123–125.MathSciNetGoogle Scholar
  4. [4]
    Floyd, E. E., On periodic maps and the Euler characteristic of associated spaces, Trans. Am. Math. Soc. 72 (1952), 138–147.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Fogarty, J., Fixed point schemes, Amer. J. Math. 95 (1973), 35–51.MathSciNetMATHCrossRefGoogle Scholar
  6. Fujita, T., On Zariski problem, Proc. Japan Acad., 55A (1979), 106–110.MATHCrossRefGoogle Scholar
  7. [7]
    Kambayashi, T. and Russell, P., On linearizing algebraic torus actions, J. Pure and Applied Algebra 23 (1982), 243–250.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Koras, M., A characterization of C 2/Z a, to appear.Google Scholar
  9. [9]
    Koras, M. and Russell, P., Gm-actions on A3, Canadian Math. Soc. Conference Proceedings, Vol. 6 (1986), 269–276.MathSciNetGoogle Scholar
  10. [10]
    Koras, M. and Russell, P., On linearizing “good” C*-actions on C 3, Canadian Math. Soc. Conference Proceedings, Vol.10 (1989), 93–101.MathSciNetGoogle Scholar
  11. [11]
    Koras, M. and Russell, P., Codimension 2 torus actions on affine n-space, Canadian Math. Soc. Conference Proceedings, Vol. 10 (1989), 103–110.MathSciNetGoogle Scholar
  12. [12]
    Koras, M. and Russell, P., Mixed C*-actions on C 3 with isolated fix-point, in preparation.Google Scholar
  13. [13]
    Kraft, H. and Schwarz, G., Reductive group actions on affine space with one-dimensional quotient, Canadian Math. Soc. Conference Proceedings, Vol.10 (1989), 125–132.MathSciNetGoogle Scholar
  14. [14]
    Kraft, H., Petrie, T. and Randall, J., Quotient varieties, Adv. Math. 74, No. 2(1989), 145–162.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Miyanishi, M. and Sugie, T., Affine surface containing cylinderlike open sets, J. Math. Kyoto Univ. 20 (1980), 11–42.MathSciNetMATHGoogle Scholar
  16. [16]
    Quillen, D., Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Russell, P., On affine-ruled rational surfaces, Math. Ann. 255 (1981), 287–302.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Suslin, A. A., On projective modules over polynomial rings, Math. USSR Sbornik 22 (1974) 595–602.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Peter Russell

There are no affiliations available

Personalised recommendations