Higher Algebraic K-Theory of Schemes and of Derived Categories

  • R. W. Thomason
  • Thomas Trobaugh
Part of the Progress in Mathematics book series (MBC, volume 88)


In this paper we prove a localization theorem for the K-theory of commutative rings and of schemes, Theorem 7.4, relating the K-groups of a scheme, of an open subscheme, and of the category of those perfect complexes on the scheme which are acyclic on the open subscheme. The localization theorem of Quillen [Q1] for K′- or G-theory is the main support of his many results on the G-theory of noetherian schemes. The previous lack of an adequate localization theorem for K-theory has obstructed development of this theory for the fifteen years since 1973. Hence our theorem unleashes a pack of new basic results hitherto known only under very restrictive hypotheses like regularity. These new results include the “Bass fundamental theorem” 6.6, the Zariski (Nisnevich) cohomolog-ical descent spectral sequence that reduces problems to the case of local (hensel local) rings 10.3 and 19.8, the Mayer-Vietoris theorem for open covers 8.1, invariance mod under polynomial extensions 9.5, Vorst-van der Kallen theory for NK 9.12, Goodwillie and Ogle-Weibel theorems relating K-theory to cyclic cohomology 9.10, mod Mayer-Vietoris for closed covers 9.8, and mod comparison between algebraic and topological K-theory 11.5 and 11.9. Indeed most known results in K-theory can be improved by the methods of this paper, by removing now unnecessary regularity, affineness, and other hypotheses.


Exact Sequence Line Bundle Spectral Sequence Abelian Category Exact Functor 
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  1. [A]
    Adams, J. F., Stable Homotopy and Generalized Homotopy, University of Chicago Press, (1974).Google Scholar
  2. [B]
    Bass, H., Algebraic K-Theory, Benjamin (1968).zbMATHGoogle Scholar
  3. [Bri]
    Brinkmann, K.-H., Diplomarbeit, Univ. Bielefeld (1978).Google Scholar
  4. [Bro]
    Brown, K., Abstract homotopy theory and generalized sheaf cohomollogy, Trans. Amer. Math. Soc. 186 (1973), 419–458.CrossRefMathSciNetGoogle Scholar
  5. [BG]
    Brown, K. and S. Gersten, Algebraic K-theory as generalized sheaf cohomology, Higher K-Theories, Springer Lect. Notes Math 341 (1973), 266–292.CrossRefMathSciNetGoogle Scholar
  6. [BK]
    Bousfield, A. K., D. M. Kan, Homotopy Limits, Completions and Localizations, Springer Lect. Notes Math. 304 (1972).CrossRefzbMATHGoogle Scholar
  7. [Ca]
    Carter, D. W., Localization in lower algebraic K-theory, Comm. Alg. 8 (1980), 603–622.CrossRefzbMATHGoogle Scholar
  8. [Co]
    Collino, A., Quittais K-theory and algebraic cycles on almost non-singular varieties. Ill. J. Math 25 (1981), 654–666.zbMATHMathSciNetGoogle Scholar
  9. [Cor]
    Cortazar, J., Rayuela. Editorial Sudamericana Sociedad Anónima (1963), translated as Hopscotch, Random House (1966).Google Scholar
  10. [DF]
    Dwyer, W., and E. Friedlander, Etale K-theory and arithmetic, Trans. Amer. Math. Soc. 292 (1985), 247–280.zbMATHMathSciNetGoogle Scholar
  11. [Gab]
    Gabber, O., K-theory of henselian local rings and henselian pairs, preprint (1985).Google Scholar
  12. [Gar]
    Gabriel, P., Des catégories abéliènnes. Bull. Soc. Math. France 90 (1962), 323–448.zbMATHMathSciNetGoogle Scholar
  13. [Ge]
    Gersten, S., The localization theorem for projective modules, Comm. Alg. 2 (1974). 307–350.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [Gil]
    Gillet, H., Comparison of K-theory spectral sequences, with applications, Algebraic A’-theory: Evanston 1980, Springer Lect. Notes Math. 854 (1981), 141–167.MathSciNetGoogle Scholar
  15. [Gi2]
    Gillet, H., Riemann-Roch theorems for higher algebraic K-theory, Adv. Math. 40 (1981). 203–289.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [Gi3]
    Gillet, H., On the K-theory of surfaces ivith multiple curves, Duke Math. J. 51(1984), 195–233.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [Gi4]
    Gillet H., The K-theory of twisted complexes. Applications of Algebraic A’-theory to Algebraic Geometry and Number Theory, Contemporary Math. 55 Pt I. Amer. Math Soc. (1986), 159–191.Google Scholar
  18. [GT]
    Gillet, H., R. W. Thomason. The K-theory of strict hensel local rings and a theorem of Sushn, J. Pure Applied Alg. 34 (1984), 241–254.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [Gir]
    Giraud, J., Cohomologie Xon Abélienne. Springer-Verlag (1971).Google Scholar
  20. [God]
    Godement R., Théorie des Faisceaux, Hermann (1964).Google Scholar
  21. [Goo]
    Goodwillie, T., Relative algebraic K-theory and cyclic homology, Ann. Math. 124 (1986), 347–402.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [Grl]
    Grayson, D., Higher algebraic K-theory II (after Quillen), Algebraic A’-theory: Evanston 1976. Springer Lect. Notes Math. 551 (1976), 217 – 240.Google Scholar
  23. [Gr2]
    Grayson, D., Exact sequences in algebraic K-theory, Ill. J. Math. 31 (1987), 598–617.zbMATHMathSciNetGoogle Scholar
  24. [Gr3]
    Grayson, D., Localization for flat modules in algebraic K-theory, J. Alg. 61 (1979), 463–496.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [Gro]
    Grothendieck, A., Sur quelques points d’algèbre homologique, Tohoku Math. J. 9 (1957), 119–221.zbMATHMathSciNetGoogle Scholar
  26. [H]
    Hartshorne, R., Residues and Duality. Springer Lect. Notes Math. 20 (1966).zbMATHGoogle Scholar
  27. [HS]
    Hinich, V., V. Shektman. Geometry of a category of complexes and algebraic K-theory, Duke Math. J. 52 (1985), 399–430.CrossRefzbMATHMathSciNetGoogle Scholar
  28. [Ja]
    Jardine, R., Stable homotopy theory of simplicial presheaves, Canad. J. Math. 39 (1987), 733–747.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [Jo]
    Jouanolou, J., Une suite exacte de Mayer-Vietoris en K-théorie algébrique, Higher K-theories, Springer Lect. Notes Math. 341 (1973), 293–316.CrossRefMathSciNetGoogle Scholar
  30. [K]
    Karoubi, M., Algèbres de Clifford et K-théorie, Ann. Sci. Ec. Norm. Sup. 1 (1968), 161–270.zbMATHMathSciNetGoogle Scholar
  31. [KS]
    Kato, K. S. Saito, Global class field theory of arithmetic schemes, Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Contemporary Math., 5 Pt. I, Amer. Math. Soc. (1986), 255–331.Google Scholar
  32. [L1]
    Levine, M., Localization on singular varieties, Invent. Math. 91 (1988), 423–464.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [L2]
    Levine, M., Bloch’s formula for singular surfaces, Topology 24 (1985), 165–174.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [Lo]
    Loday, J.-L., Cyclic homology: A survey, Banach Center Publications (Warsaw) 18 (1986), 285–307.MathSciNetGoogle Scholar
  35. [Ma1]
    May, J. P., Simplicial Objects in Algebraic Topology, van Nostrand (1967).Google Scholar
  36. [Ma2]
    May, J. P., Pairings of categories and spectra, J. Pure Applied Alg. 19 (1980), 259–282.CrossRefGoogle Scholar
  37. [Mum]
    Mumford, D., Lectures on Curves on an Algebraic Surface, Princeton Univ. Press (1966).zbMATHGoogle Scholar
  38. [N1]
    Nisnevich, Y., Arithmetic and cohomological invariants of semi-simple group schemes and compactifications of locally symmetric spaces, Funct. Anal. Appl., 14 no 1, (1980), 75–76.MathSciNetGoogle Scholar
  39. [N2]
    Nisnevich, Y., Adeles and Grothendieck topologies, preprint (1982).Google Scholar
  40. [N3]
    Nisnevich, Y., The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, Algebraic K-Theory: Connections with Geometry and Topology, Kluwer (1989), 241–341.CrossRefGoogle Scholar
  41. [O]
    Ogle, C., On the K-theory and cyclic homology of a square-zero ideal, J. Pure Applied Alg. 46 (1987), 233–248.CrossRefzbMATHMathSciNetGoogle Scholar
  42. [OW]
    Ogle, C., C. Weibel, Relative K-theory and cyclic homology, to appear.Google Scholar
  43. [PW]
    Pedrini, C., C. Weibel, K-theory and Chow groups on singular varieties, Applications of Algebraic K-theory, Contemp. Math. 55 Pt 1 (1986), 339–370.CrossRefMathSciNetGoogle Scholar
  44. [Q1]
    Quillen, D., Higher algebraic K-theory I, Higher K-theories, Springer Lect. Notes Math. 341 (1973), 85–147.CrossRefMathSciNetGoogle Scholar
  45. [Q2]
    Quillen, D., Homotopical Algebra, Springer Lect. Notes Math. 43 (1967).Google Scholar
  46. [Se]
    Serre, J. -P., Prolongement des faisceaux analytiques cohérents, Ann. Inst. Fourier 16 (1966), 363–374.CrossRefzbMATHGoogle Scholar
  47. [Sta]
    Staffeldt, R., On fundamental theorems of algebraic K-theory, K-Theory 2 (1989), 511–532.CrossRefzbMATHMathSciNetGoogle Scholar
  48. [Su1]
    Suslin, A., On the K-theory of algebraically closed fields, Invent. Math. 73 (1983), 241–245.CrossRefzbMATHMathSciNetGoogle Scholar
  49. [Su2]
    Suslin, A., On the K-theory of local fields, J. Pure Applied Alg. 34 (1984), 301–318.CrossRefzbMATHMathSciNetGoogle Scholar
  50. [Sw]
    Swan, R., K-theory of quadric hypersurfaces, Ann. Math. 122 (1985), 113–153.CrossRefzbMATHMathSciNetGoogle Scholar
  51. [Th1]
    Thomason, R. W., Algebraic K- theory and étale cohomology, Ann. Sci. Ec. Norm. Sup. 18 (1985), 437–552.zbMATHMathSciNetGoogle Scholar
  52. [Th2]
    Thomason, R. W., Homotopy colimits in the category of small categories, Math. Proc. Camb, Philos. Soc. 85 (1979), 91–109.CrossRefzbMATHMathSciNetGoogle Scholar
  53. [Th3]
    Thomason, R. W., First quadrant spectral sequences in algebraic K-theory via homotopy colimits, Comm. Alg. 15 (1982), 1589–1668.CrossRefMathSciNetGoogle Scholar
  54. [Th4]
    Thomason, R. W., Lefschetz-Riemann-Roch theorem and coherent trace formula, Invent. Math. 85 (1986), 515–543.CrossRefzbMATHMathSciNetGoogle Scholar
  55. [vdK]
    van der Kallen, W., Descent for K-theory of polynomial rings, Math. Zeit. 191 (1986), 405–415.CrossRefzbMATHGoogle Scholar
  56. [V]
    Verdier, J.-L., Catégories dérivées, pp. 262–311 of [SGA 4½], see below.Google Scholar
  57. [Vo]
    Vorst, T., Localization of the K-theory of polynomial extensions, Math. Ann. 244 (1979), 333–353.Google Scholar
  58. [W]
    Waldhausen, F., Algebraic K-theory of spaces, Algebraic and Geometric Topology, Springer Lect. Notes Math. 1126 (1985), 318–419.CrossRefMathSciNetGoogle Scholar
  59. [We1]
    Weibel, C., Homotopy algebraic K-theory, Contemp. Math. 83 (1989), 461–488.CrossRefMathSciNetGoogle Scholar
  60. [We2]
    Weibel, C., Mayer-Vietoris sequences and module structures on NK, Algebraic K-theory: Evanston 1980, Springer Lect. Notes Math. 854 (1981), 466–493.CrossRefMathSciNetGoogle Scholar
  61. [We3]
    Weibel, C., Mayer- Vietoris sequences and modp K-theory, Algebraic K-theory: Oberwolfach 1980, Springer Lect. Notes Math. 966 (1982), 390–407.CrossRefMathSciNetGoogle Scholar
  62. [We4]
    Weibel, C., Negative K-theory of varieties with isolated singularities, J. Pure Applied Alg. 34 (1984), 331–342.CrossRefzbMATHMathSciNetGoogle Scholar
  63. [We5]
    Weibel, C., A Brown-G ersten spectral sequence for the K-theory of varieties with isolated singularities, Adv. Math 73 (1989), 192–203.CrossRefzbMATHMathSciNetGoogle Scholar
  64. [We6]
    Weibel, C., Module structures on the K-theory of graded rings, J. Alg. 105 (1987), 465–483.CrossRefzbMATHMathSciNetGoogle Scholar
  65. [EGA]
    Grothendieck, A., J. Dieudonné, Éléments de géométrie algébrique, Publ. Math. I. H. E. S. Nos. 8, 11, 17, 20, 24, 28, 32 (1961–1967), Grundleheren, 166 (1971), Press. Univ. France, Springer-Verlag.zbMATHGoogle Scholar
  66. [SGA 1]
    Grothendieck, A., Revetments étales et groupe fondamental, Springer Lect. Notes Math. 224 (1971).Google Scholar
  67. [SGA 2]
    Grothendieck, A., Cohomologie local des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, North-Holland (1968).Google Scholar
  68. [SGA 4]
    Artin, M., A. Grothendieck, J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, Springer Lect. Notes Math., 269, 270, 305 (1972–1973).MathSciNetGoogle Scholar
  69. [SGA 4½]
    Deligne, P. et al, Cohomologie étale, Springer Lect. Notes Math. 569 (1977).CrossRefzbMATHGoogle Scholar
  70. [SGA 6]
    Berthelot, P., A. Grothendieck, L. Illusie, Théorie des intersections et théorème de Riemann-Roch, Springer Lect. Notes Math. 225 (1971).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • R. W. Thomason
    • 1
  • Thomas Trobaugh
    • 2
  1. 1.Department of MathematicsThe Johns Hopkins UniversityUSA
  2. 2.Université de Paris-SudOrsay CedexFrance

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