Selecta Mathematica

, Volume 24, Issue 2, pp 1039–1091 | Cite as

The index map in algebraic K-theory

  • Oliver Braunling
  • Michael Groechenig
  • Jesse Wolfson
Article
  • 37 Downloads

Abstract

In this paper we provide a detailed description of the K-theory torsor constructed by S. Saito for a Tate R-module, and its analogue for general idempotent complete exact categories. We study the classifying map of this torsor in detail, construct an explicit simplicial model, and link it to the index theory of Fredholm operators. The torsor is also related to canonical central extensions of loop groups. More precisely, we compare the K-theory torsor to previously studied dimension and determinant torsors.

Mathematics Subject Classification

Primary 19D55 Secondary 19K56 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Artin, M., Mazur, B.: Etale Homotopy, Lecture Notes in Mathematics, vol. 100. Springer, Berlin (1969)CrossRefMATHGoogle Scholar
  2. 2.
    Atiyah, M.: \(K\)-theory, second edn. In: Advanced Book Classics. Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA (1989). Notes by D. W. AndersonGoogle Scholar
  3. 3.
    Baranovsky, V.: Uhlenbeck compactification as a functor. Int. Math. Res. Not. 2015(23), 12678–12712 (2015)MathSciNetMATHGoogle Scholar
  4. 4.
    Beilinson, A., Bloch, S., Esnault, H.: \(\epsilon \)-factors for Gauss–Manin determinants. Mosc. Math. J. 2(3), 477–532 (2002). Dedicated to Yuri I. Manin on the occasion of his 65th birthdayMathSciNetMATHGoogle Scholar
  5. 5.
    Beilinson, A: How to glue perverse sheaves. In: \(K\)-Theory, Arithmetic and Geometry Moscow, 1984–1986, Lecture Notes in Mathematics, vol. 1289, pp. 42–51. Springer, Berlin (1987)Google Scholar
  6. 6.
    Braunling, O., Groechenig, M., Wolfson, J.: The \({A}_{\infty }\)-structure of the index map (2016). https://jpwolfson.files.wordpress.com/2013/02/segal.pdf
  7. 7.
    Braunling, O., Groechenig, M., Wolfson, J.: Tate objects in exact categories. Mosc. Math. J. 16(3), 433–504 (2016). With an appendix by Jan Šťovíček and Jan TrlifajMathSciNetMATHGoogle Scholar
  8. 8.
    Bühler, T.: Exact categories. Expo. Math. 28(1), 1–69 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chinburg, T., Pappas, G., Taylor, M.J.: Higher adeles and non-abelian Riemann–Roch. Adv. Math. 281, 928–1024 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Douglas, R.G.: Banach algebra techniques in operator theory, 2 edn. In: Graduate Texts in Mathematics, vol. 179. Springer, New York (1998)Google Scholar
  11. 11.
    Drinfeld, V.: Infinite-dimensional vector bundles in algebraic geometry: an introduction. In: The Unity of Mathematics, Progress in Mathematics, vol. 244, pp. 263–304. Birkhäuser, Boston (2006)Google Scholar
  12. 12.
    Duskin, J.W.: Simplicial matrices and the nerves of weak \(n\)-categories I: nerves of bicategories. Theory Appl. Categ. 9, 198–308 (2001)MathSciNetMATHGoogle Scholar
  13. 13.
    Gersten, S.: Higher \(K\)-theory of rings. Algebraic \(K\)-theory, I: Higher \(K\)-theories (Proceedings of the Conference Seattle Research Center, Battelle Memorial Institute 1972), Lecture Notes in Mathematics, Vol. 341, pp. 3–42. Springer, Berlin (1973)Google Scholar
  14. 14.
    Goerss, P., Jardine, J.F.: Simplicial Homotopy Theory. Birkhäuser-Verlag, Basel (2009)CrossRefMATHGoogle Scholar
  15. 15.
    Hennion, B.: Tate objects in stable \((\infty ,1)\)-categories. Homol Homotopy Appl. arXiv:1606.05527
  16. 16.
    Jänich, K.: Vektorraumbündel und der Raum der Fredholm-Operatoren. Math. Ann. 161, 129–142 (1965)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kan, D.M.: On c.s.s. complexes. Am. J. Math. 79, 449–476 (1957)CrossRefGoogle Scholar
  18. 18.
    Kapranov, M.: Semiinfinite symmetric powers. arXiv:math/0107089
  19. 19.
    Kapranov, M.: Letter to Brylinski, unpublished (1995)Google Scholar
  20. 20.
    Kapranov, M.: Double affine Hecke algebras and 2-dimensional local fields. J. Am. Math. Soc. 14(1), 239–262 (2001). (electronic)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kato, K.: Existence theorem for higher local fields. In: Invitation to Higher Local Fields (Münster, 1999), Geometry and Topology Monographs, vol. 3, pp. 165–195. Geometry and Topology publication, Coventry (2000)Google Scholar
  22. 22.
    Keller, Bernhard: On the cyclic homology of exact categories. J. Pure Appl. Algebra 136(1), 1–56 (1999)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kontsevich, M.: Lecture at Orsay (7 December 1995)Google Scholar
  24. 24.
    Kapranov, M., Vasserot, E.: Vertex algebras and the formal loop space. Publ. Math. Inst. Hautes Études Sci. 100, 206–269 (2004)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lefschetz, S.: Algebraic topology. In: American Mathematical Society Colloquium Publications, vol. 27. American Mathematical Society, New York (1942)Google Scholar
  26. 26.
    Lurie, J.: Higher topos theory. Annals of Mathematics Studies, 170, Princeton University Press, New Jersey (2009)Google Scholar
  27. 27.
    Nikolaus, T., Schreiber, U., Stevenson, D.: Principal \(\infty \)-bundles: general theory. J. Homotopy Relat. Struct. 10(4), 749–801 (2015)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Previdi, L.: Locally compact objects in exact categories. Int. J. Math. 22(12), 1787–1821 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Pressley, A., Segal, G.: Loop groups. In: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford Science Publications, New York (1986)Google Scholar
  30. 30.
    Quillen, D.: Higher algebraic \(K\)-theory. I. Algebraic \(K\)-theory, I: higher \(K\)-theories (Proceedings of the Conference Battelle Memorial Institute, Seattle, Washington, 1972), Lecture Notes in Mathematics, Vol. 341, pp. 85–147. Springer, Berlin (1973)Google Scholar
  31. 31.
    Saito, S.: Higher Tate central extensions via \(K\)-theory and infinity-topos theory. 05 (2014). arXiv:1405.0923
  32. 32.
    Saito, S.: On Previdi’s delooping conjecture for \(K\)-theory. Algebra Number Theory 9(1), 1–11 (2015)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Schlichting, M.: Delooping the \(K\)-theory of exact categories. Topology 43(5), 1089–1103 (2004)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Schlichting, M.: Negative \(K\)-theory of derived categories. Math. Z. 253(1), 97–134 (2006)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Sato, M., Sato, Y.: Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold. In: Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982). North-Holland Mathematics Studies, vol. 81, pp. 259–271. North-Holland, Amsterdam (1983)Google Scholar
  36. 36.
    Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. Inst. Hautes Études Sci. 61, 5–65 (1985)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Thomason, R.W., Trobaugh, T.: Higher algebraic \(K\)-theory of schemes and of derived categories. In: The Grothendieck Festschrift, Vol. III, Progress in Mathematics, vol. 88, pp. 247–435. Birkhäuser, Boston (1990)Google Scholar
  38. 38.
    Waldhausen, F.: Algebraic \(K\)-theory of spaces. In: Algebraic Geometry and Topology (New Brunswick, N.J, 1983), Lecture Notes in Mathematics, vol. 1126. Springer (1985)Google Scholar
  39. 39.
    Weibel, C.: The \(K\)-Book: An Introduction to Algebraic \(K\)-Theory. American Mathematical Society, Providence (2013)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Oliver Braunling
    • 1
  • Michael Groechenig
    • 2
  • Jesse Wolfson
    • 3
  1. 1.Department of MathematicsUniversität FreiburgFreiburgGermany
  2. 2.Department of MathematicsFreie Universität BerlinBerlinGermany
  3. 3.Department of MathematicsUniversity of California, IrvineIrvineUSA

Personalised recommendations