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K-Theory of Truncated Polynomial Algebras

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Handbook of K-Theory

Abstract

In general, if A is a ring and I ⊂ A a two-sided ideal, one defines the K-theory of A relative to I to be the mapping fiber of the map of K-theory spectra induced by the canonical projection from A to A/I. Hence, there is anatural exact triangle of spectra

$$ K(A,I) \rightarrow K(A) \rightarrow K(A/I) \xrightarrow{\partial} K(A,I)[-1] $$

and an induced natural long-exact sequence of K-groups

$$ ... \rightarrow K_{q}(A,I) \rightarrow K_{q}(A) \rightarrow K_{q}(A/I) \xrightarrow{\partial} K_{q-1}(A,I) \rightarrow ... $$

.

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References

  1. J. F. Adams, Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture, Algebraic topology, Aarhus 1982 (Aarhus, 1982), Lecture Notes in Math., vol. 1051, Springer-Verlag, New York, 1984, pp. 483–532.

    Google Scholar 

  2. J. E. Aisbett, E. Lluis-Puebla, and V. P. Snaith, On K ∗(Z| n) and K ∗(Fq[t]|(t2 )). With an appendix by Christophe Soulé, Mem. Amer. Math. Soc., vol. 57, Amer. Math. Soc., Providence, RI, 1985.

    Google Scholar 

  3. P. Berthelot, Cohomologie cristalline des schémas de caractéristique p > 0, Lecture Notes in Math., vol. 407, Springer-Verlag, New York, 1974.

    MATH  Google Scholar 

  4. S. Betley and C. Schlichtkrull, The cyclotomic trace and K -theoretical curves, Topology, (to appear).

    Google Scholar 

  5. S. Bloch, Algebraic K -theory and crystalline cohomology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 187–268.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Bökstedt, Topological Hochschild homology, Preprint, Bielefeld University, 1985.

    Google Scholar 

  7. M. Bökstedt, W.-C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic K -theory of spaces, Invent. Math. 111 (1993), 465–540.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257–281.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. K. Bousfield and D. M. Kan,Homotopylimits, completions and localizations, Lecture Notes in Math., vol. 304, Springer-Verlag, New York, 1972.

    Google Scholar 

  10. L. Breen, Extensions du groupe additif, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 39–125.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Brun, Topological Hochschild homology of Z| pn , J. Pure Appl. Algebra 148 (2000), 29–76.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Brun, Filtered topological cyclic homology and relative K -theory of nilpotent ideals, Algebr. Geom. Topol. 1 (2001), 201–230.

    Article  MATH  MathSciNet  Google Scholar 

  13. C. Carathéodory, Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193–217.

    Article  MATH  Google Scholar 

  14. A. Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257–360.

    Article  MathSciNet  Google Scholar 

  15. L. Evens and E. M. Friedlander, On K ∗(Z| p2 Z) and related homology groups, Trans. Amer. Math. Soc. 270 (1982), 1–46.

    Article  MATH  MathSciNet  Google Scholar 

  16. P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35, Springer-Verlag, New York, 1967.

    MATH  Google Scholar 

  17. D. Gale, Neighborly and cyclic polytopes, Convexity (Seattle, Washington, 1961), Proc. Symp. Pure Math., vol. 7, Amer. Math. Soc., Providence, RI, 1963, pp. 225–232.

    Google Scholar 

  18. T. Geisser, On K3 of Witt vectors of length two over finite fields, K-Theory 12 (1997), 193–226.

    Article  MATH  MathSciNet  Google Scholar 

  19. T. Geisser and L. Hesselholt, The de Rham-Witt complex and p -adic vanishing cycles, Preprint 2003.

    Google Scholar 

  20. T. Geisser and L. Hesselholt, Topological cyclic homology of schemes, K-theory (Seattle, 1997), Proc. Symp. Pure Math., vol. 67, 1999, pp. 41–87.

    Google Scholar 

  21. T. G. Goodwillie, Relative algebraic K -theory and cyclic homology, Ann. of Math. 124 (1986), 347–402.

    Article  MathSciNet  Google Scholar 

  22. T. G. Goodwillie, Calculus I: The first derivative of pseudoisotopy theory, K-theory 4 (1990), 1–27.

    Article  MATH  MathSciNet  Google Scholar 

  23. T. G. Goodwillie, Calculus II: Analytic functors, K-theory 5 (1992), 295–332.

    Article  MATH  Google Scholar 

  24. J. A. Guccione, J. J. Guccione, M. J. Redondo, A. Solotar, and O. E. Villamayor, Cyclic homology of algebras with one generator, K-theory 5 (1991), 51–68.

    Article  MATH  MathSciNet  Google Scholar 

  25. L. Hesselholt, On the p -typical curves in Quillen’s K -theory, Acta Math. 177 (1997), 1–53.

    Article  MathSciNet  Google Scholar 

  26. L. Hesselholt, Algebraic K -theory and trace invariants, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 415–425.

    Google Scholar 

  27. L. Hesselholt and I. Madsen, On the de Rham-Witt complex in mixed characteristic, Ann. Sci. École Norm. Sup., 37(4) (2004), 1–43.

    MATH  MathSciNet  Google Scholar 

  28. L. Hesselholt and I. Madsen, Cyclic polytopes and the K -theory of truncated polynomial algebras, Invent. Math. 130 (1997), 73–97.

    Article  MATH  MathSciNet  Google Scholar 

  29. L. Hesselholt and I. Madsen, On the K -theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), 29–102.

    Article  MATH  MathSciNet  Google Scholar 

  30. L. Hesselholt and I. Madsen, On the K -theory of nilpotent endomorphisms, Homotopy methods in algebraic topology (Boulder, CO, 1999), Contemp. Math., vol. 271, Amer. Math. Soc., Providence, RI, 2001, pp. 127–140.

    Google Scholar 

  31. L. Hesselholt and I. Madsen, On the K -theory of local fields, Ann. of Math. 158 (2003), 1–113.

    Article  MATH  MathSciNet  Google Scholar 

  32. G. Hochschild, B. Kostant, and A. Rosenberg, Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383–408.

    Article  MATH  MathSciNet  Google Scholar 

  33. M. Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 156 (2001), 63–127.

    Article  MathSciNet  Google Scholar 

  34. M. Hovey, B. Shipley, and J. Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149–208.

    Article  MATH  MathSciNet  Google Scholar 

  35. L. Illusie, Complexe de de Rham–Witt et cohomologie cristalline, Ann. Scient. Éc. Norm. Sup. (4) 12 (1979), 501–661.

    MATH  MathSciNet  Google Scholar 

  36. M. Jibladze and T. Pirashvili, Cohomology of algebraic theories, J. Algebra 137 (1991), 253–296.

    Article  MATH  MathSciNet  Google Scholar 

  37. J. D. S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987), 403–423.

    Article  MATH  MathSciNet  Google Scholar 

  38. C. Kassel, Cyclic homology, comodules, and mixed complexes, J. Algebra 107 (1987), 195–216.

    Article  MATH  MathSciNet  Google Scholar 

  39. K. Kato, A generalization of local class field theory by using K -groups. II, J. Fac. Sci. Univ. Tokyo Sect. I. Math. 27 (1980), 603–683.

    MATH  Google Scholar 

  40. L. G. Lewis, J. P. May, and M. Steinberger, Equivariant stable homotopy theory, Lecture Notes in Math., vol. 1213, Springer-Verlag, New York, 1986.

    MATH  Google Scholar 

  41. J.-L. Loday, Cyclic homology, Grundlehren der mathematischen Wissenschaften, vol. 301, Springer-Verlag, New York, 1992.

    MATH  Google Scholar 

  42. J.-L. Loday and D. Quillen, Cyclic homology and the Lie algebra homology of matrices, Comm. Math. Helv. 59 (1984), 565–591.

    Article  MATH  MathSciNet  Google Scholar 

  43. S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1971.

    Google Scholar 

  44. I. Madsen, Algebraic K -theory and traces, Current Developments in Mathematics, 1995, International Press, Cambridge, MA, 1996, pp. 191–321.

    Google Scholar 

  45. M. A. Mandell and J. P. May, Equivariant orthogonal spectra and S -modules, Mem. Amer. Math. Soc., vol. 159, Amer. Math. Soc., Providence, RI, 2002.

    Google Scholar 

  46. T. Masuda and T. Natsume, Cyclic cohomology of certain affine schemes, Publ. Res. Inst. Math. Sci. 21 (1985), 1261–1279.

    Article  MATH  MathSciNet  Google Scholar 

  47. R. McCarthy, Relative algebraic K -theory and topological cyclic homology, Acta Math. 179 (1997), 197–222.

    Article  MATH  MathSciNet  Google Scholar 

  48. I. A. Panin, On a theorem of Hurewicz and K -theory of complete discrete valuation rings, Math. USS. Izvestiya 29 (1987), 119–131.

    Article  MATH  Google Scholar 

  49. T. Pirashvili and F. Waldhausen, Mac Lane homology and topological Hochschild homology, J. Pure Appl. Algebra 82 (1992), 81–98.

    Article  MATH  MathSciNet  Google Scholar 

  50. D. Popescu, General Néron desingularisation and approximation (Letter to the editor), Nagoya Math. J. 118 (1990), 45–53.

    MATH  MathSciNet  Google Scholar 

  51. D. Quillen, Higher algebraic K -theory I, Algebraic K-theory I: Higher K-theories (Battelle Memorial Inst., Seattle, Washington, 1972), Lecture Notes in Math., vol. 341, Springer-Verlag, New York, 1973.

    Google Scholar 

  52. E. H. Spanier, Algebraic topology. Corrected reprint, Springer-Verlag, New York, 1981.

    Google Scholar 

  53. A. A. Suslin, On the K -theory of local fields, J. Pure Appl. Alg. 34 (1984), 304–318.

    Article  MathSciNet  Google Scholar 

  54. R. G. Swan, Néron–Popescu desingularization, Algebra and geometry (Taipei, 1995), Lect. Algebra Geom., 2, Internat. Press, Cambridge, MA, 1998, pp. 135–192.

    Google Scholar 

  55. B. L. Tsygan, Homology of matrix algebras over rings and the Hochschild homology, Uspekhi Mat. Nauk. 38 (1983), 217–218.

    MATH  MathSciNet  Google Scholar 

  56. W. van der Kallen, Le K2 des nombres duaux, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A1204–A1207.

    Google Scholar 

  57. C. A. Weibel, Mayer-Vietoris sequences and module structure on NK ∗, AlgebraicK-theory (Evanston, IL, 1980), Lecture Notes in Math., vol. 854, Springer-Verlag, New York, 1981.

    Google Scholar 

  58. C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

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

    Lars Hesselholt

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  1. Lars Hesselholt
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Northwestern University, 60208, Evanston, IL, USA

    Eric M. Friedlander

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, 61801, Urbana, IL, USA

    Daniel R. Grayson

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Hesselholt, L. (2005). K-Theory of Truncated Polynomial Algebras. In: Friedlander, E., Grayson, D. (eds) Handbook of K-Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27855-9_3

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