Skip to main content
Log in

Vector bundles over affine surfaces

  • Published:
Inventiones Mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Abhyankar, S.S.: Resolution of singularities of embedded algebraic surfaces. New York: Academic Press 1966

    MATH  Google Scholar 

  2. Andreotti, A., Frankel, T.: The Lefschetz theorem of hyperplane sections. Ann. of Math.69, 713–717 (1959)

    Article  MathSciNet  Google Scholar 

  3. Artin, M., Bloch, L., Kas, A., Lieberman, D.: Zero cycles on surfaces withp g =0. To appear

  4. Barsotti, I.: Structure theorems for group varieties. Ann. di Mat.38, 77–119 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bass, H.: AlgebraicK-theory. New York: Benjamin 1968

    Google Scholar 

  6. Bass, H.: Libération des modules projectifs sur certains anneaux de polynômes. Sem. Bourb. 26° an. 1973/74 no 448

  7. Bloch, S.:K 2 of artinianQ-algebras with application to algebraic cycles. To appear

  8. Bombieri, E.: The pluricanonical map of a complex surface. Several complex variables. Maryland 1970, p. 35–87. Lecture Notes in Mathematics155. Berlin-Heidelberg-New York: Sprigner 1970

    Book  Google Scholar 

  9. Borel, A.: Linear algebraic groups. New York: Benjamin 1969

    MATH  Google Scholar 

  10. Borel, S., Serre, J.-P.: Groupes de Lie et puissances reduit de Steenrod. Am. J. Math.75, 409–448 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  11. Borel, A., Serre, J.-P.: Le théorème de Riemann-Roch. Bull. Soc. Math. (France)86, 97–136 (1958)

    MathSciNet  MATH  Google Scholar 

  12. Bott, R.: The space of loops on a Lie group. Mich. Math. J.5, 35–61 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eagon, J.: The Grothendieck group of finitely generated modules. Proc. Amer. Math. Soc.19, 1341–1345 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eklof, P.: Lefschetz's principle and local functors. Proc. Amer. Math. Soc.37, 333–339 (1973)

    MathSciNet  MATH  Google Scholar 

  15. Fulton, W.: Rational equivalence on singular varieties. To appear

  16. Goldman, O.: Determinants in projective modules. Nagoya Math. J.18, 27–36 (1961)

    MathSciNet  MATH  Google Scholar 

  17. Goodman, J.E.: Affine open subsets of algebraic varieties and ample divisors. Ann. of Math.89, 160–183 (1969)

    Article  MathSciNet  Google Scholar 

  18. Grauert, H.: Analytische Faserungen über holomorphvollständigen Räumen. Math. Ann.135, 263–273 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grothendieck, A.: La théorie des classes de Chern. Bull. Soc. Math. (France)86, 137–154 (1958)

    MathSciNet  MATH  Google Scholar 

  20. Grothendieck, A.: Revêtements etales et groupe fondamental (SGAI) Lect. Notes in Math. 224. Berlin-Heidelberg-New York: Springer Verlag 1971

    Google Scholar 

  21. Grothendieck, A., Berthelot, P., Illusie, L., et al.: Théorie des intersections et thèorème de Riemann-Roch (SGA6). Lect. Notes in Math.225. Berlin-Heidelberg-New York: Springer 1971

    Google Scholar 

  22. Gunning, R., Rossi, H.: Analytic functions of several complex variables. Prentice Hall Englewood Cliffs, New Jersey 1965

    MATH  Google Scholar 

  23. Hartshorne, R.: Ample Subvarieties of Algebraic Varieties, Lecture Notes in Mathematics156. Berlin-Heidelberg-New York: Springer 1970

    Book  MATH  Google Scholar 

  24. Hartshorne, R.: Equivalence Relations on Algebraic Cycles, Lecture Notes, Summer Inst. on Alg. Geom. at Humboldt State Univ. Amer. Math. Soc. 1974

  25. Hu, S. T.: Homotopy Theory, New York: Academic Press 1959

    MATH  Google Scholar 

  26. Kaplansky, I.: Commutative Rings, Boston: Allyn and Bacon 1970

    MATH  Google Scholar 

  27. Kaplansky, I.: Fields and rings. Chicago 1969

  28. Lang, S.: On quasi algebraic closure. Ann. of Math.55, 373–390 (1952)

    Article  MathSciNet  Google Scholar 

  29. Lang, S.: Diophantine geometry, New York: Interscience 1962

    MATH  Google Scholar 

  30. Lang, S.: Abelian varieties. New York: Interscience 1959

    Google Scholar 

  31. Lech, C.: A note on recurring series. Ark. Math.2, 417–421 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lutz, E.: Sur l'équationy 2=x 3Ax−B dans les corpsp-adiques. J. Math.177, 238–247 (1937)

    Google Scholar 

  33. Milnor, J.: Construction of universal bundles II. Ann. of Math.63, 430–436 (1956)

    Article  MathSciNet  Google Scholar 

  34. Milnor, J.: Morse theory. Ann. Math. Studies 51, Princeton 1963

  35. Mumford, D.: Abelian varieties, Oxford 1970

  36. Mumford, D.: Rational equivalence of 0-cycles on surfaces. J. Math. Kyoto Univ.9, 195–204 (1969)

    MathSciNet  MATH  Google Scholar 

  37. Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publ. Math. I.H.E.S.9, 5–22 (1961)

    MathSciNet  MATH  Google Scholar 

  38. Murthy, M.P.: Vector bundles over affine surfaces birationally equivalent to a ruled surface. Ann. of Math.89, 242–253 (1969)

    Article  MathSciNet  Google Scholar 

  39. Narasimhan, R.: On the homology groups of Stein spaces. Inv. Math.2, 377–385 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  40. Raynaud, M.: Modules projectifs universels. Inv. Math.6, 1–26 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  41. Roitman, A.A.: On τ-equivalence of zero-dimensional cycles. Math. USSR Sbornik15, 555–567 (1971)

    Article  MATH  Google Scholar 

  42. Roitman, A.A.: Rational equivalence of zero-cycles. Math. USSR Sbornik18, 571–588 (1972)

    Article  Google Scholar 

  43. Roitman, A.A.: Ruledness of an algebraic surface with a finite-dimensional non-zero group of classes of zero-dimensional cycles of degree zero modulo rational equivalence. Func. Anal. and Appl.8, 82–83 (1974)

    Article  MathSciNet  Google Scholar 

  44. Rosenlicht, M.: Some basic theorems on algebraic groups. Am. J. Math.78, 401–443 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  45. Šafarevič, J.: Algebraic surfaces. Proc. Steklov Inst. Math. 75 (1965)

  46. Šafarevič, J.: Lectures on minimal models and birational transformations of two-dimensional schemes. Bombay: Tata Institute 1966

    Google Scholar 

  47. Schwarzenberger, R.: Vector bundles on the projective plane. Proc. London Math. Soc.11, 623–640 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  48. Serre, J.P.: Homologie singulière des espaces fibres. Applications. Ann. of Math.54, 425–505 (1951)

    Article  MathSciNet  Google Scholar 

  49. Serre, J-P.: Groupes d'homotopie et classes de groupes abéliens. Ann of Math.58, 258–294 (1953)

    Article  MathSciNet  Google Scholar 

  50. Serre, J-P.: Faisceaux algébriques cohérents. Ann. of Math.61, 197–278 (1955)

    Article  MathSciNet  Google Scholar 

  51. Serre, J-P.: Algèbre locale-multiplicités. Lect. Notes in Math.11. Berlin-Heidelberg-New York: Springer 1965

    MATH  Google Scholar 

  52. Serre, J-P.: Corps locaux. Paris: Hermann 1962

    MATH  Google Scholar 

  53. Serre, J-P.: Groups algébriques et corps classes. Paris: Hermann 1959

    Google Scholar 

  54. Serre, J-P.: Modules projectifs et espaces fibres à fibre vectorielle. Sem. Dubreil-Pisot 11. Paris 1957/8

  55. Serre, J.-P.: Sur les modules projectifs. Sem. Dubreil-Pisot 14 (1960-(61), No.2

  56. Serre, J-P.: Classes des corps cyclotomique, Sem. Bourbaki 174 (1958)

  57. Swan, R.G.: Vector bundles and projective modules. Trans. Amer. Math. Soc.105, 264–277 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  58. Swan, R.G.: AlgebraicK-theory. Lect. Notes in Math.76 Berlin-Heidelberg-New York: Springer 1968

    Google Scholar 

  59. Swan, R.G.:K-Theory of finite groups and orders. Notes. by E. G. Evans. Lecture Notes in Mathematics 149. Berlin-Heidelberg-New York Springer 1970

    Book  Google Scholar 

  60. Swan, R. G.: A cancellation theorem for projective modules in the metastable range. Inventiones math.27, 23–43 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  61. Swan, R.G., Towber, J.: A class of projective modules which are nearly free. J. Alg.36, 427–434 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  62. Tsen, C.: Divisionalgebren über Funktionenkörper. Nachr. Ges. Wiss. Göttingen 1933, 335–7 (1975)

  63. Weil, A.: Foundations of algebraic geometry. Amer. Math. Soc. Colloq. Publ. XXIX, Providence 1962

  64. Zariski, O.: Algebraic surfaces. Berlin-Heidelberg-New York: Springer 1971

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

To Jean-Pierre Serre

Rights and permissions

Reprints and permissions

About this article

Cite this article

Murthy, M.P., Swan, R.G. Vector bundles over affine surfaces. Invent Math 36, 125–165 (1976). https://doi.org/10.1007/BF01390007

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01390007

Keywords

Navigation