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The Unity of Mathematics pp 263–304Cite as

Infinite-Dimensional Vector Bundles in Algebraic Geometry

An Introduction

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Part of the book series: Progress in Mathematics ((PM,volume 244))

Abstract

The goal of this work is to show that there is a reasonable algebro-geometric notion of vector bundle with infinite-dimensional locally linearly compact fibers and that these objects appear “in nature.” Our approach is based on some results and ideas discovered in algebra during the period 1958–1972 by H. Bass, L. Gruson, I. Kaplansky, M. Karoubi, and M. Raynaud.

To Izrail Moiseevich Gelfand with deepest gratitude and admiration.

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References

  1. E. Arbarello, C. De Concini, and V. G. Kac, Infinite wedge representation and the reciprocity law on algebraic curves, in Theta Functions, Bowdoin 1987, Proceedings of Symposia in Pure Mathematics 49, Part 1, American Mathematical Society, Providence, RI, 1989, 171–190.

    Google Scholar 

  2. M. Artin and B. Mazur, Etale Homotopy, Lecture Notes in Mathematics 100, Springer-Verlag, Berlin, New York, Heidelberg, 1969.

    MATH  Google Scholar 

  3. H. Bass, Algebraic K-Theory, Benjamin, New York, 1968.

    MATH  Google Scholar 

  4. H. Bass, Big projective modules are free, Illinois J. Math., 7 (1963), 24–31.

    MATH  MathSciNet  Google Scholar 

  5. A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, in Analyse et topologie sur les espaces singuliers, Vol. 1, Astérisque 100, Société Mathématique de France, Paris, 1982.

    Google Scholar 

  6. A. Beilinson, S. Bloch, and H. Esnault, Epsilon-factors for Gauss-Manin determinants, Moscow Math, J., 2-3 (2002), 477–532; see also xxx.lanl.gov, e-print math.AG/0111277.

    MATH  MathSciNet  Google Scholar 

  7. V. G. Berkovich. Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs 33, American Mathematical Society, Providence, RI, 1990.

    MATH  Google Scholar 

  8. A. Braverman, M. Finkelberg, and D. Gaitsgory, Uhlenbeck spaces via affine Lie algebras, e-print AG/0301176, 2003.

    Google Scholar 

  9. S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry, Grundlehren der Mathematischen Wissenschaften 261, Springer-Verlag, Berlin, 1984.

    MATH  Google Scholar 

  10. J.-L. Brylinski, Central extensions and reciprocity laws, Cahiers Topol. Géom. Différentielle Catég., 38-3 (1997), 193–215.

    MATH  MathSciNet  Google Scholar 

  11. P. Deligne, La formule de dualité globale, in Theorie des topos et cohomologie étale des schemas (SGA 4), Tome 3, Lecture Notes in Mathematics 305, Springer-Verlag, Berlin, New York, Heidelberg, 1973, 481–587.

    Google Scholar 

  12. P. Deligne, Le déterminant de la cohomologie, in K. A. Ribet, ed., Current Trends in Arithmetical Algebraic Geometry, Contemporary Mathematics 67, American Mathematical Society, Providence, RI, 1987, 93–177.

    Google Scholar 

  13. V. Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, to appear at xxx.lanl.gov.

    Google Scholar 

  14. T. Ekedahl, On the adic formalism, in Grothendieck Festschrift, Vol. II, Progress in Mathematics 87, Birkhäuser Boston, Cambridge, MA, 1990, 197–218.

    Google Scholar 

  15. M. Finkelberg, D. Gaitsgory, and A. Kuznetsov, Uhlenbeck spaces for \( \mathbb{A}^2 \) and affine Lie algebra \( \widehat{\mathfrak{s}\mathfrak{l}}_n \), Publ. Res. Inst. Math. Sci., 39-4 (2003), 721–766.

    Article  MATH  MathSciNet  Google Scholar 

  16. P. Freyd, Abelian Categories: An Introduction to the Theory of Functors, Harper and Row, New York, 1964.

    MATH  Google Scholar 

  17. V. E. Govorov, On flat modules, Siberian Math. J., 6 (1965), 300–304 (in Russian).

    MATH  MathSciNet  Google Scholar 

  18. A. Grothendieck, Éléments de géométrie algébrique IV: Étude locale des schémas et de morphismes de schémas, première partie, Publications Mathématiques IHES 20, Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France, 1964.

    Google Scholar 

  19. A. Grothendieck, Éléments de géométrie algébrique IV: Étude locale des schémas et de morphismes de schémas, quatrième partie, Publications Mathématiques IHES 32, Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France, 1967.

    Google Scholar 

  20. A. Grothendieck, Catégories cofibrées additives et complexe cotangent relatif, Lecture Notes in Mathematics 79, Springer-Verlag, Berlin, New York, Heidelberg, 1968.

    MATH  Google Scholar 

  21. A. Grothendieck and J. L. Verdier, Préfaisceaux, in Théorie des topos et cohomologie étale des schemas (SGA 4), Tome 1, Lecture Notes in Mathematics 269, Springer-Verlag, Berlin, New York, Heidelberg, 1972, 1–217.

    Google Scholar 

  22. U. Jannsen, Continuous étale cohomology, Math. Ann., 280-2 (1988), 207–245.

    Article  MATH  MathSciNet  Google Scholar 

  23. I. Kaplansky, Projective modules, Ann. Math., 68 (1958), 372–377.

    Article  MathSciNet  Google Scholar 

  24. I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, Ann Arbor, MI, 1969.

    MATH  Google Scholar 

  25. M. Kapranov, Semiinfinite symmetric powers, e-print math.QA/0107089, 2001.

    Google Scholar 

  26. M. Kapranov and E. Vasserot, Vertex algebras and the formal loop space, Publ. Math. IHES, 100 (2004), 209–269.

    MATH  MathSciNet  Google Scholar 

  27. D. Lazard, Autour de la platitude, Bull. Soc. Math. France, 97 (1969), 81–128.

    MATH  MathSciNet  Google Scholar 

  28. S. Lefschetz, Algebraic Topology, AMS Colloquium Publications 27, American Mathematical Society, Providence, RI, 1942.

    MATH  Google Scholar 

  29. E. Looijenga, Motivic measures, in Seminaire Bourbaki, Vol. 1999/2000, Astérisque 276, Société Mathématique de France, Paris, 2002, 267–297; see also xxx.lanl.gov, e-print math.AG/0006220.

    Google Scholar 

  30. F. Malikov, V. Schechtman, and A. Vaintrob, Chiral de Rham complex, Comm. Math. Phys., 204 (1999), 439–473.

    Article  MATH  MathSciNet  Google Scholar 

  31. J. P. May, E ∞ spaces, group completions, and permutative categories, in New Developments in Topology, London Mathematical Society Lecture Note Series 11, Cambridge University Press, London, 1974, 61–93.

    Google Scholar 

  32. Pure Imaginary Observer, The future of field theory, J. Irreproducible Results, 12-3 (1963), 3–5 (English); in Fiziki Prodolzhayut Shutit, Mir, Moscow, 1968 (Russian).

    Google Scholar 

  33. A. Pressley and G. Segal, Loop Groups, Oxford University Press, Oxford, UK, 1986.

    MATH  Google Scholar 

  34. L. Reid, N-dimensional rings with an isolated singular point having nonzero K −N , K-Theory, 1-2 (1987), 197–205.

    Article  MATH  MathSciNet  Google Scholar 

  35. M. Raynaud and L. Gruson, Critères de platitude et de projectivité, Invent. Math., 13 (1971), 1–89.

    Article  MATH  MathSciNet  Google Scholar 

  36. G. Segal, Categories and cohomology theories, Topology, 13 (1974), 293–312.

    Article  MATH  MathSciNet  Google Scholar 

  37. J.-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, in Algèbre Théorie Nombres, Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot 11, 1957–1958, Secrétariat Mathématique, Paris, 1958, Exposé 23.

    Google Scholar 

  38. R. M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. Math. (2), 92 (1970), 1–56.

    Article  MathSciNet  Google Scholar 

  39. J. Tate, Residues of differentials on curves, Ann. Sci. École Norm. Sup. Ser. 4, 1 (1968), 149–159.

    MATH  MathSciNet  Google Scholar 

  40. R. W. Thomason and T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, in Grothendieck Festschrift,Vol. III, Progress in Mathematics 88, Birkhäuser Boston, Cambridge, MA, 1990, 247–435.

    Google Scholar 

  41. V. Voevodsky, \( \mathbb{A} \) 1-homotopy theory, Documenta Math., Extra Vol. I (1998) (Proceedings of the International Congress of Mathematicians (Berlin, 1998), Vol. I), 579–604 (electronic).

    MathSciNet  Google Scholar 

  42. V. Voevodsky, A. Suslin, and E. M. Friedlander, Cycles, Transfers, and Motivic Homology Theories, Annals of Mathematics Studies 143, Princeton University Press, Princeton, NJ, 2000.

    MATH  Google Scholar 

  43. V. Voevodsky, C. Mazza, and C. Weibel, Lectures on motivic cohomology, available online from http://math.rutgers.edu/~weibel/motiviclectures.html.

    Google Scholar 

  44. C. Weibel, Negative K-theory of varieties with isolated singularities, J. Pure Appl. Algebra, 34-2–3 (1984), 331–342.

    Article  MATH  MathSciNet  Google Scholar 

  45. C. Weibel, Pic is a contracted functor, Invent. Math., 103-2 (1991), 351–377.

    Article  MATH  MathSciNet  Google Scholar 

  46. C. Weibel, Negative K-theory of surfaces, Duke Math. J., 108-1 (2001), 1–35.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL, 60637, USA

    Vladimir Drinfeld

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  1. Vladimir Drinfeld
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Editors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Pavel Etingof  & I. M. Singer  & 

  2. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Vladimir Retakh

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Drinfeld, V. (2006). Infinite-Dimensional Vector Bundles in Algebraic Geometry. In: Etingof, P., Retakh, V., Singer, I.M. (eds) The Unity of Mathematics. Progress in Mathematics, vol 244. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4467-9_7

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