Skip to main content
SpringerLink
Log in

The index map in algebraic K-theory

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Artin, M., Mazur, B.: Etale Homotopy, Lecture Notes in Mathematics, vol. 100. Springer, Berlin (1969)

    Book  MATH  Google Scholar 

  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. Anderson

  3. Baranovsky, V.: Uhlenbeck compactification as a functor. Int. Math. Res. Not. 2015(23), 12678–12712 (2015)

    MathSciNet  MATH  Google Scholar 

  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 birthday

    MathSciNet  MATH  Google Scholar 

  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)

  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. 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 Trlifaj

    MathSciNet  MATH  Google Scholar 

  8. Bühler, T.: Exact categories. Expo. Math. 28(1), 1–69 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chinburg, T., Pappas, G., Taylor, M.J.: Higher adeles and non-abelian Riemann–Roch. Adv. Math. 281, 928–1024 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Douglas, R.G.: Banach algebra techniques in operator theory, 2 edn. In: Graduate Texts in Mathematics, vol. 179. Springer, New York (1998)

  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)

  12. Duskin, J.W.: Simplicial matrices and the nerves of weak \(n\)-categories I: nerves of bicategories. Theory Appl. Categ. 9, 198–308 (2001)

    MathSciNet  MATH  Google Scholar 

  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)

  14. Goerss, P., Jardine, J.F.: Simplicial Homotopy Theory. Birkhäuser-Verlag, Basel (2009)

    Book  MATH  Google Scholar 

  15. Hennion, B.: Tate objects in stable \((\infty ,1)\)-categories. Homol Homotopy Appl. arXiv:1606.05527

  16. Jänich, K.: Vektorraumbündel und der Raum der Fredholm-Operatoren. Math. Ann. 161, 129–142 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kan, D.M.: On c.s.s. complexes. Am. J. Math. 79, 449–476 (1957)

    Article  Google Scholar 

  18. Kapranov, M.: Semiinfinite symmetric powers. arXiv:math/0107089

  19. Kapranov, M.: Letter to Brylinski, unpublished (1995)

  20. Kapranov, M.: Double affine Hecke algebras and 2-dimensional local fields. J. Am. Math. Soc. 14(1), 239–262 (2001). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  22. Keller, Bernhard: On the cyclic homology of exact categories. J. Pure Appl. Algebra 136(1), 1–56 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kontsevich, M.: Lecture at Orsay (7 December 1995)

  24. Kapranov, M., Vasserot, E.: Vertex algebras and the formal loop space. Publ. Math. Inst. Hautes Études Sci. 100, 206–269 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lefschetz, S.: Algebraic topology. In: American Mathematical Society Colloquium Publications, vol. 27. American Mathematical Society, New York (1942)

  26. Lurie, J.: Higher topos theory. Annals of Mathematics Studies, 170, Princeton University Press, New Jersey (2009)

  27. Nikolaus, T., Schreiber, U., Stevenson, D.: Principal \(\infty \)-bundles: general theory. J. Homotopy Relat. Struct. 10(4), 749–801 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Previdi, L.: Locally compact objects in exact categories. Int. J. Math. 22(12), 1787–1821 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pressley, A., Segal, G.: Loop groups. In: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford Science Publications, New York (1986)

  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)

  31. Saito, S.: Higher Tate central extensions via \(K\)-theory and infinity-topos theory. 05 (2014). arXiv:1405.0923

  32. Saito, S.: On Previdi’s delooping conjecture for \(K\)-theory. Algebra Number Theory 9(1), 1–11 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Schlichting, M.: Delooping the \(K\)-theory of exact categories. Topology 43(5), 1089–1103 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Schlichting, M.: Negative \(K\)-theory of derived categories. Math. Z. 253(1), 97–134 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  36. Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. Inst. Hautes Études Sci. 61, 5–65 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  39. Weibel, C.: The \(K\)-Book: An Introduction to Algebraic \(K\)-Theory. American Mathematical Society, Providence (2013)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Universität Freiburg, Freiburg, Germany

    Oliver Braunling

  2. Department of Mathematics, Freie Universität Berlin, Berlin, Germany

    Michael Groechenig

  3. Department of Mathematics, University of California, Irvine, Irvine, CA, USA

    Jesse Wolfson

Authors
  1. Oliver Braunling
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Groechenig
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jesse Wolfson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jesse Wolfson.

Additional information

O.B. was supported by DFG SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties” and Alexander von Humboldt Foundation. M.G. was supported by EPRSC Grant No. EP/G06170X/1. J.W. was partially supported by an NSF Graduate Research Fellowship under Grant No. DGE-0824162, by an NSF Research Training Group in the Mathematical Sciences under Grant No. DMS-0636646, and by an NSF Post-doctoral Research Fellowship under Grant No. DMS-1400349. He was a guest of K. Saito at IPMU while this paper was being completed. Our research was supported in part by NSF Grant No. DMS-1303100 and EPSRC Mathematics Platform Grant EP/I019111/1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Braunling, O., Groechenig, M. & Wolfson, J. The index map in algebraic K-theory. Sel. Math. New Ser. 24, 1039–1091 (2018). https://doi.org/10.1007/s00029-017-0364-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-017-0364-0

Mathematics Subject Classification

Search

Navigation