Algebraic K-Theory

  • Bjørn Ian Dundas
  • Thomas G. Goodwillie
  • Randy McCarthy
Part of the Algebra and Applications book series (AA, volume 18)

Abstract

This chapter begins with a nontechnical overview of algebraic K-theory, including some historical motivation and its development. This is followed by the introduction of Waldhausen’s S-construction and proofs of its essential first properties. Finally, algebraic K-theory is compared with the homology of categories, providing a first model for the “differential” of algebraic K-theory.

Keywords

Filtration Manifold 

References

  1. 1.
    Théorie des Intersections et Théorème de Riemann-Roch, volume 225 of Lecture Notes in Mathematics. Springer, Berlin, 1971. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J.P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J.P. Serre. Google Scholar
  2. 9.
    M.F. Atiyah. K-Theory, 2nd edition, Advanced Book Classics. Addison-Wesley, Redwood City, 1989. Notes by D.W. Anderson. Google Scholar
  3. 12.
    M. Barratt and S. Priddy. On the homology of non-connected monoids and their associated groups. Comment. Math. Helv., 47:1–14, 1972. MathSciNetMATHCrossRefGoogle Scholar
  4. 13.
    H. Bass. Algebraic K-Theory. Benjamin, New York, 1968. Google Scholar
  5. 14.
    H. Bass. Personal reminiscences of the birth of algebraic K-theory. K-Theory, 30(3):203–209, 2003. Special issue in honor of Hyman Bass on his seventieth birthday, Part III. MathSciNetMATHCrossRefGoogle Scholar
  6. 16.
    A.J. Berrick. An Approach to Algebraic K-Theory, volume 56 of Research Notes in Mathematics. Pitman, Boston, 1982. Google Scholar
  7. 34.
    A. Borel. Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. (4), 7(1975):235–272, 1974. MathSciNetMATHGoogle Scholar
  8. 35.
    A. Borel and J.-P. Serre. Le théorème de Riemann-Roch. Bull. Soc. Math. Fr., 86:97–136, 1958. MathSciNetMATHGoogle Scholar
  9. 36.
    R. Bott. The periodicity theorem for the classical groups and some of its applications. Adv. Math., 4(1970):353–411, 1970. MathSciNetMATHCrossRefGoogle Scholar
  10. 40.
    A.K. Bousfield and D.M. Kan. Homotopy Limits, Completions and Localizations, volume 304 of Lecture Notes in Mathematics Springer, Berlin, 1972. MATHCrossRefGoogle Scholar
  11. 54.
    P.M. Cohn. Some remarks on the invariant basis property. Topology, 5:215–228, 1966. MathSciNetMATHCrossRefGoogle Scholar
  12. 60.
    R.K. Dennis and M.R. Stein. K 2 of discrete valuation rings. Adv. Math., 18(2):182–238, 1975. MathSciNetMATHCrossRefGoogle Scholar
  13. 79.
    P. Elbaz-Vincent, H. Gangl, and C. Soulé. Quelques calculs de la cohomologie de GLN(ℤ) et de la K-théorie de ℤ. C. R. Math. Acad. Sci. Paris, 335(4):321–324, 2002. MathSciNetMATHCrossRefGoogle Scholar
  14. 81.
    F.T. Farrell and L.E. Jones. Rigidity in geometry and topology. In Proceedings of the International Congress of Mathematicians, Vol. I, II, Kyoto, 1990, pages 653–663. Math. Soc. Japan, Tokyo, 1991. Google Scholar
  15. 86.
    E.M. Friedlander and D.R. Grayson, editors. Handbook of K-Theory, Vols. 1, 2. Springer, Berlin, 2005. Google Scholar
  16. 87.
    E.M. Friedlander and B. Mazur. Filtrations on the homology of algebraic varieties. Mem. Am. Math. Soc., 110(529):110, 1994. With an appendix by Daniel Quillen. MathSciNetGoogle Scholar
  17. 88.
    O. Gabber. K-theory of Henselian local rings and Henselian pairs. In Algebraic K-Theory, Commutative Algebra, and Algebraic Geometry, Santa Margherita Ligure, 1989, volume 126 of Contemporary Mathematics, pages 59–70. Am. Math. Soc., Providence, 1992. Google Scholar
  18. 91.
    S. Galatius, U. Tillmann, I. Madsen, and M. Weiss. The homotopy type of the cobordism category. Acta Math., 202(2):195–239, 2009. MathSciNetMATHCrossRefGoogle Scholar
  19. 96.
    S.M. Gersten. K 3 of a ring is H 3 of the Steinberg group. Proc. Am. Math. Soc., 37:366–368, 1973. MathSciNetMATHGoogle Scholar
  20. 97.
    S.M. Gersten. K-theory of free rings. Commun. Algebra, 1:39–64, 1974. MathSciNetMATHCrossRefGoogle Scholar
  21. 98.
    H. Gillet and D.R. Grayson. The loop space of the Q-construction. Ill. J. Math., 31(4):574–597, 1987. MathSciNetMATHGoogle Scholar
  22. 101.
    T.G. Goodwillie. On the general linear group and Hochschild homology. Ann. Math. (2), 121(2):383–407, 1985. MathSciNetMATHCrossRefGoogle Scholar
  23. 102.
    T.G. Goodwillie. Relative algebraic K-theory and cyclic homology. Ann. Math. (2), 124(2):347–402, 1986. MathSciNetMATHCrossRefGoogle Scholar
  24. 108.
    D. Grayson. Higher algebraic K-theory. II (after Daniel Quillen). In Algebraic K-Theory, Proc. Conf., Northwestern Univ., Evanston, IL, 1976, volume 551 of Lecture Notes in Mathematics, pages 217–240. Springer, Berlin, 1976. CrossRefGoogle Scholar
  25. 109.
    D.R. Grayson. The K-theory of endomorphisms. J. Algebra, 48(2):439–446, 1977. MathSciNetMATHCrossRefGoogle Scholar
  26. 110.
    D.R. Grayson. Exact sequences in algebraic K-theory. Ill. J. Math., 31(4):598–617, 1987. MathSciNetMATHGoogle Scholar
  27. 111.
    D.R. Grayson. Exterior power operations on higher K-theory. K-Theory, 3(3):247–260, 1989. MathSciNetMATHCrossRefGoogle Scholar
  28. 112.
    D.R. Grayson. On the K-theory of fields. In Algebraic K-Theory and Algebraic Number Theory, Honolulu, HI, 1987, volume 83 of Contemporary Mathematics, pages 31–55. Amer. Math. Soc., Providence, 1989. Google Scholar
  29. 113.
    D.R. Grayson. Adams operations on higher K-theory. K-Theory, 6(2):97–111, 1992. MathSciNetMATHCrossRefGoogle Scholar
  30. 119.
    B. Harris. Bott periodicity via simplicial spaces. J. Algebra, 62(2):450–454, 1980. MathSciNetMATHCrossRefGoogle Scholar
  31. 122.
    J.-C. Hausmann and D. Husemoller. Acyclic maps. Enseign. Math. (2), 25(1–2):53–75, 1979. MathSciNetMATHGoogle Scholar
  32. 133.
    G. Higman. The units of group-rings. Proc. Lond. Math. Soc. (2), 46:231–248, 1940. MathSciNetCrossRefGoogle Scholar
  33. 134.
    H.L. Hiller. λ-rings and algebraic K-theory. J. Pure Appl. Algebra, 20(3):241–266, 1981. MathSciNetMATHCrossRefGoogle Scholar
  34. 139.
    M. Hovey. Model Categories, volume 63 of Mathematical Surveys and Monographs. Am. Math. Soc., Providence, 1999. MATHGoogle Scholar
  35. 147.
    H. Inassaridze. Algebraic K-Theory, volume 311 of Mathematics and Its Applications. Kluwer Academic, Dordrecht, 1995. Google Scholar
  36. 154.
    D.M. Kan and W.P. Thurston. Every connected space has the homology of a K(π,1). Topology, 15(3):253–258, 1976. MathSciNetMATHCrossRefGoogle Scholar
  37. 159.
    C. Kassel. K-théorie relative d’un idéal bilatère de carré nul: étude homologique en basse dimension. In Algebraic K-Theory, Proc. Conf., Northwestern Univ., Evanston, IL, 1980, volume 854 of Lecture Notes in Mathematics, pages 249–261. Springer, Berlin, 1981. Google Scholar
  38. 160.
    C. Kassel. La K-théorie stable. Bull. Soc. Math. Fr., 110(4):381–416, 1982. MathSciNetMATHGoogle Scholar
  39. 163.
    M.A. Kervaire. Le théorème de Barden-Mazur-Stallings. Comment. Math. Helv., 40:31–42, 1965. MathSciNetMATHCrossRefGoogle Scholar
  40. 164.
    M.A. Kervaire. Smooth homology spheres and their fundamental groups. Trans. Am. Math. Soc., 144:67–72, 1969. MathSciNetMATHCrossRefGoogle Scholar
  41. 165.
    M.A. Kervaire. Multiplicateurs de Schur et K-théorie. In Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), pages 212–225. Springer, New York, 1970. Google Scholar
  42. 167.
    Ch. Kratzer. λ-structure en K-théorie algébrique. Comment. Math. Helv., 55(2):233–254, 1980. MathSciNetMATHCrossRefGoogle Scholar
  43. 171.
    W.G. Leavitt. Modules without invariant basis number. Proc. Am. Math. Soc., 8:322–328, 1957. MathSciNetMATHCrossRefGoogle Scholar
  44. 172.
    R. Lee and R.H. Szczarba. The group K 3(Z) is cyclic of order forty-eight. Ann. Math. (2), 104(1):31–60, 1976. MathSciNetMATHCrossRefGoogle Scholar
  45. 175.
    S. Lichtenbaum. Values of zeta-functions, étale cohomology, and algebraic K-theory. In Algebraic K-Theory, II: “Classical” Algebraic K-Theory and Connections with Arithmetic, Proc. Conf., Battelle Memorial Inst., Seattle, WA, 1972, volume 342 of Lecture Notes in Mathematics, pages 489–501. Springer, Berlin, 1973. Google Scholar
  46. 178.
    J.-L. Loday. Applications algébriques du tore dans la sphère et de S p×S q dans S p+q. In Algebraic K-Theory, II: “Classical” Algebraic K-Theory and Connections with Arithmetic, Proc. Conf., Seattle Res. Center, Battelle Memorial Inst., 1972, volume 342 of Lecture Notes in Mathematics, pages 79–91. Springer, Berlin, 1973. CrossRefGoogle Scholar
  47. 179.
    J.-L. Loday. K-théorie algébrique et représentations de groupes. Ann. Sci. École Norm. Sup. (4), 9(3):309–377, 1976. MathSciNetMATHGoogle Scholar
  48. 189.
    H. Maazen and J. Stienstra. A presentation for K 2 of split radical pairs. J. Pure Appl. Algebra, 10(3):271–294, 1977. MathSciNetCrossRefGoogle Scholar
  49. 191.
    S. Mac Lane. Categories for the Working Mathematician, 2nd edition, volume 5 of Graduate Texts in Mathematics. Springer, New York, 1998. MATHGoogle Scholar
  50. 193.
    I. Madsen and M. Weiss. The stable mapping class group and stable homotopy theory. In European Congress of Mathematics, pages 283–307. Eur. Math. Soc., Zürich, 2005. Google Scholar
  51. 194.
    B.A. Magurn, editor. Reviews in K-Theory, 1940–1984. Am. Math. Soc., Providence, 1985. Reviews reprinted from Mathematical Reviews. Google Scholar
  52. 202.
    B. Mazur. Relative neighborhoods and the theorems of Smale. Ann. Math. (2), 77:232–249, 1963. MathSciNetMATHCrossRefGoogle Scholar
  53. 203.
    R. McCarthy. On fundamental theorems of algebraic K-theory. Topology, 32(2):325–328, 1993. MathSciNetMATHCrossRefGoogle Scholar
  54. 210.
    J. Milnor. Whitehead torsion. Bull. Am. Math. Soc., 72:358–426, 1966. MathSciNetMATHCrossRefGoogle Scholar
  55. 212.
    J. Milnor. Two complexes which are homeomorphic but combinatorially distinct. Ann. Math. (2), 74:575–590, 1961. MathSciNetMATHCrossRefGoogle Scholar
  56. 213.
    J. Milnor. Introduction to Algebraic K-Theory, volume 72 of Annals of Mathematics Studies. Princeton University Press, Princeton, 1971. Google Scholar
  57. 214.
    J.W. Milnor and J.D. Stasheff. Characteristic Classes, volume 76 of Annals of Mathematics Studies. Princeton University Press, Princeton, 1974. MATHGoogle Scholar
  58. 215.
    B. Mitchell. Rings with several objects. Adv. Math., 8:1–161, 1972. MATHCrossRefGoogle Scholar
  59. 216.
    S.A. Mitchell. On the Lichtenbaum-Quillen conjectures from a stable homotopy-theoretic viewpoint. In Algebraic Topology and Its Applications, volume 27 of Mathematical Sciences Research Institute Publications, pages 163–240. Springer, New York, 1994. CrossRefGoogle Scholar
  60. 218.
    D. Mumford. Towards an enumerative geometry of the moduli space of curves. In Arithmetic and Geometry, Vol. II, volume 36 of Progress of Mathematics, pages 271–328. Birkhäuser, Boston, 1983. Google Scholar
  61. 220.
    R. Oliver. Whitehead Groups of Finite Groups, volume 132 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1988. MATHCrossRefGoogle Scholar
  62. 228.
    D. Quillen. Cohomology of groups. In Actes du Congrès International des Mathématiciens, Tome 2, Nice, 1970, pages 47–51. Gauthier-Villars, Paris, 1971. Google Scholar
  63. 229.
    D. Quillen. Letter to Graeme Segal, July 25 1972. Google Scholar
  64. 231.
    D. Quillen. Finite generation of the groups K i of rings of algebraic integers. In Algebraic K-Theory, I: Higher K-Theories, Proc. Conf., Battelle Memorial Inst., Seattle, WA, 1972, volume 341 of Lecture Notes in Mathematics, pages 179–198. Springer, Berlin, 1973. Google Scholar
  65. 232.
    D. Quillen. Higher algebraic K-theory. I. In Algebraic K-Theory, I: Higher K-Theories, Proc. Conf., Battelle Memorial Inst., Seattle, WA, 1972, volume 341 of Lecture Notes in Mathematics, pages 85–147. Springer, Berlin, 1973. Google Scholar
  66. 236.
    A.A. Ranicki, A.J. Casson, D.P. Sullivan, M.A. Armstrong, C.P. Rourke, and G.E. Cooke. The Hauptvermutung Book, volume 1 of K-Monographs in Mathematics. Kluwer Academic, Dordrecht, 1996. A collection of papers of the topology of manifolds. MATHGoogle Scholar
  67. 239.
    J. Rognes and C. Weibel. Two-primary algebraic K-theory of rings of integers in number fields. J. Am. Math. Soc., 13(1):1–54, 2000. Appendix A by Manfred Kolster. MathSciNetMATHCrossRefGoogle Scholar
  68. 241.
    J. Rognes. K 4(Z) is the trivial group. Topology, 39(2):267–281, 2000. MathSciNetMATHCrossRefGoogle Scholar
  69. 244.
    J. Rosenberg. Algebraic K-Theory and Its Applications, volume 147 of Graduate Texts in Mathematics. Springer, New York, 1994. CrossRefGoogle Scholar
  70. 245.
    M. Schlichting. Negative K-theory of derived categories. Math. Z., 253(1):97–134, 2006. MathSciNetMATHCrossRefGoogle Scholar
  71. 247.
    C. Schlichtkrull. Units of ring spectra and their traces in algebraic K-theory. Geom. Topol., 8:645–673, 2004 (electronic). MathSciNetMATHCrossRefGoogle Scholar
  72. 258.
    J.-P. Serre. Faisceaux algébriques cohérents. Ann. Math. (2), 61:197–278, 1955. MATHCrossRefGoogle Scholar
  73. 264.
    S. Smale. On the structure of manifolds. Am. J. Math., 84:387–399, 1962. MathSciNetMATHCrossRefGoogle Scholar
  74. 270.
    V. Srinivas. Algebraic K-Theory, 2nd edition, volume 90 of Progress in Mathematics. Birkhäuser, Boston, 1996. Google Scholar
  75. 271.
    R.E. Staffeldt. On fundamental theorems of algebraic K-theory. K-Theory, 2(4):511–532, 1989. MathSciNetMATHCrossRefGoogle Scholar
  76. 272.
    J.R. Stallings. On infinite processes leading to differentiability in the complement of a point. In Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pages 245–254. Princeton University Press, Princeton, 1965. Google Scholar
  77. 273.
    A. Suslin. On the K-theory of algebraically closed fields. Invent. Math., 73(2):241–245, 1983. MathSciNetMATHCrossRefGoogle Scholar
  78. 277.
    A.A. Suslin. Algebraic K-theory of fields. In Proceedings of the International Congress of Mathematicians, Vols. 1, 2, Berkeley, CA, 1986, pages 222–244. Amer. Math. Soc., Providence, 1987. Google Scholar
  79. 280.
    R.G. Swan. Vector bundles and projective modules. Trans. Am. Math. Soc., 105:264–277, 1962. MathSciNetMATHCrossRefGoogle Scholar
  80. 281.
    R.G. Swan. Excision in algebraic K-theory. J. Pure Appl. Algebra, 1(3):221–252, 1971. MathSciNetMATHCrossRefGoogle Scholar
  81. 284.
    R.W. Thomason and T. Trobaugh. Higher algebraic K-theory of schemes and of derived categories. In The Grothendieck Festschrift, Vol. III, volume 88 of Progress of Mathematics, pages 247–435. Birkhäuser, Boston, 1990. CrossRefGoogle Scholar
  82. 285.
    U. Tillmann. On the homotopy of the stable mapping class group. Invent. Math., 130(2):257–275, 1997. MathSciNetMATHCrossRefGoogle Scholar
  83. 292.
    W. van der Kallen, H. Maazen, and J. Stienstra. A presentation for some K 2(n,R). Bull. Am. Math. Soc., 81(5):934–936, 1975. MATHCrossRefGoogle Scholar
  84. 293.
    V. Voevodsky. Motivic cohomology with Z/2-coefficients. Publ. Math. Inst. Hautes Études Sci., 98:59–104, 2003. MathSciNetMATHCrossRefGoogle Scholar
  85. 297.
    F. Waldhausen. Algebraic K-theory of generalized free products, I, II. Ann. Math. (2), 108(1):135–204, 1978. MathSciNetMATHCrossRefGoogle Scholar
  86. 299.
    F. Waldhausen. Algebraic K-theory of topological spaces, II. In Algebraic Topology, Proc. Sympos., Univ. Aarhus, Aarhus, 1978, volume 763 of Lecture Notes in Mathematics, pages 356–394. Springer, Berlin, 1979. Google Scholar
  87. 301.
    F. Waldhausen. Algebraic K-theory of spaces. In Algebraic and Geometric Topology, New Brunswick, NJ, 1983, volume 1126 of Lecture Notes in Mathematics, pages 318–419. Springer, Berlin, 1985. CrossRefGoogle Scholar
  88. 302.
    F. Waldhausen. Algebraic K-theory of spaces, concordance, and stable homotopy theory. In Algebraic Topology and Algebraic K-Theory, Princeton, NJ, 1983, volume 113 of Annals of Mathematics Studies, pages 392–417. Princeton University Press, Princeton, 1987. Google Scholar
  89. 306.
    C. Weibel. An introduction to algebraic K-theory. In progress, some of it is on the web, http://math.rutgers.edu/~weibel/Kbook.html.
  90. 309.
    C. Weibel. Algebraic K-theory of rings of integers in local and global fields. In Handbook of K-Theory, Vols. 1, 2, pages 139–190. Springer, Berlin, 2005. Google Scholar
  91. 310.
    C.A. Weibel. The development of algebraic K-theory before 1980. In Algebra, K-Theory, Groups, and Education, New York, 1997, volume 243 of Contemporary Mathematics, pages 211–238. Am. Math. Soc., Providence, 1999. CrossRefGoogle Scholar
  92. 314.
    J.H.C. Whitehead. Simplicial spaces, nuclei and m-groups. Proc. Lond. Math. Soc. (2), 45:243–327, 1939. CrossRefGoogle Scholar
  93. 315.
    J.H.C. Whitehead. On incidence matrices, nuclei and homotopy types. Ann. Math. (2), 42:1197–1239, 1941. MathSciNetMATHCrossRefGoogle Scholar
  94. 316.
    J.H.C. Whitehead. Simple homotopy types. Am. J. Math., 72:1–57, 1950. MathSciNetMATHCrossRefGoogle Scholar
  95. 317.
    D. Yao. A devissage theorem in Waldhausen K-theory. J. Algebra, 176(3):755–761, 1995. MathSciNetMATHCrossRefGoogle Scholar
  96. 318.
    D. Yau. Lambda-Rings. World Scientific, Hackensack, 2010. MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Bjørn Ian Dundas
    • 1
  • Thomas G. Goodwillie
    • 2
  • Randy McCarthy
    • 3
  1. 1.Department of MathematicsUniversity of BergenBergenNorway
  2. 2.Mathematics DepartmentBrown UniversityProvidenceUSA
  3. 3.Department of MathematicsUniversity of IllinoisUrbanaUSA

Personalised recommendations