Topics in Algebraic and Topological K-Theory pp 167-241

Part of the Lecture Notes in Mathematics book series (LNM, volume 2008)

Higher Algebraic K-Theory (After Quillen, Thomason and Others)



We present an introduction (with a few proofs) to higher algebraic K-theory of schemes based on the work of Quillen, Waldhausen, Thomason and others. Our emphasis is on the application of triangulated category methods in algebraic K-theory.


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  1. 1.
    Adams, J.F.: Stable homotopy and generalised homology. In: Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1974)Google Scholar
  2. 2.
    Mandell, M.A., Blumberg, A.J.: Localization theorems in topological Hochschild homology and topological cyclic homology.arXiv:0802.3938v2 (2008)Google Scholar
  3. 3.
    Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading (1969)MATHGoogle Scholar
  4. 4.
    Alonso Tarrío, L., JeremíasLópez, A., Lipman, J.: Local homology and cohomology on schemes. Ann. Sci. École Norm. Sup. (4), 30(1), 1–39 (1997)Google Scholar
  5. 5.
    Alonso Tarrío, L., JeremíasLópez, A., SoutoSalorio, M.J.: Localization in categories of complexes and unbounded resolutions. Can. J. Math. 52(2), 225–247 (2000)Google Scholar
  6. 6.
    Balmer, P.: Triangular Witt groups. I. The 12-term localization exact sequence. K-Theory 19(4), 311–363 (2000)Google Scholar
  7. 7.
    Bass, H.: Algebraic K-theory. W.A. Benjamin, New York (1968)Google Scholar
  8. 8.
    Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and topology on singular spaces, I (Luminy, 1981). Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris (1982)Google Scholar
  9. 9.
    Bousfield, A.K., Friedlander, E.M.: Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. In: Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II. Lecture Notes in Mathematics, vol. 658, pp. 80–130. Springer, Berlin (1978)Google Scholar
  10. 10.
    Bousfield, A.K., Kan, D.M.: Homotopy limits, completions and localizations. In: Lecture Notes in Mathematics, vol. 304. Springer, Berlin (1972)Google Scholar
  11. 11.
    Bloch, S., Kato, K.: p-adic étale cohomology. Inst. Hautes Études Sci. Publ. Math. (63), 107–152 (1986)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bondal, A.I., Larsen, M., Lunts, V.A.: Grothendieck ring of pretriangulated categories. Int. Math. Res. Not. 29, 1461–1495 (2004)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Bloch, S.: Algebraic cycles and higher K-theory. Adv. Math. 61(3), 267–304 (1986)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bökstedt, M., Neeman, A.: Homotopy limits in triangulated categories. Compos. Math. 86(2), 209–234 (1993)MATHGoogle Scholar
  15. 15.
    Borelli, M.: Divisorial varieties. Pac. J. Math. 13, 375–388 (1963)MATHMathSciNetGoogle Scholar
  16. 16.
    Balmer, P., Schlichting, M.: Idempotent completion of triangulated categories. J.Algebra 236(2), 819–834 (2001)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Buehler, T.: Exact categories. Expositiones Mathematicae, 28(1), 1–69 (2010)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Bondal, A., vanden Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36, 258 (2003)MATHMathSciNetGoogle Scholar
  19. 19.
    Cortiñas, G., Haesemeyer, C., Schlichting, M., Weibel, C.: Cyclic homology, cdh-cohomology and negative K-theory. Ann. Math. (2) 167(2), 549–573 (2008)MATHGoogle Scholar
  20. 20.
    Cortiñas, G., Haesemeyer, C., Weibel, C.: K-regularity, cdh-fibrant Hochschild homology, and a conjecture of Vorst. J. Am. Math. Soc. 21(2), 547–561 (2008)MATHCrossRefGoogle Scholar
  21. 21.
    Cortiñas, G.: Algebraic v. topological K-theory: a friendly match. In this volumeGoogle Scholar
  22. 22.
    Cortiñas, G.: The obstruction to excision in K-theory and in cyclic homology. Invent. Math. 164(1), 143–173 (2006)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Drinfeld, V.: DG quotients of DG categories. J. Algebra 272(2), 643–691 (2004)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Friedlander, E.M., Grayson, D.R. (eds.): Handbook of K-theory, vol. 1,2. Springer, Berlin (2005)Google Scholar
  25. 25.
    Fritsch, R., Piccinini, R.A.: Cellular structures in topology. In: Cambridge Studies in Advanced Mathematics, vol. 19. Cambridge University Press, Cambridge (1990)Google Scholar
  26. 26.
    Franke, J.: On the Brown representability theorem for triangulated categories. Topology 40(4), 667–680 (2001)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Friedlander, E.M., Suslin, A.: The spectral sequence relating algebraic K-theory to motivic cohomology. Ann. Sci. École Norm. Sup. (4) 35(6), 773–875 (2002)Google Scholar
  28. 28.
    Fulton, W.: Intersection theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], second edn., vol.2. Springer, Berlin (1998)Google Scholar
  29. 29.
    Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. France 90, 323–448 (1962)MATHMathSciNetGoogle Scholar
  30. 30.
    Geisser, T., Hesselholt, L.: On the vanishing of negative k-groups. arXiv:0811.0652 (2008)Google Scholar
  31. 31.
    Goerss, P.G., Jardine, J.F.: Simplicial homotopy theory. In: Simplicial homotopy theory, vol. 174. Birkhäuser, Basel (1999)Google Scholar
  32. 32.
    Geisser, T., Levine, M.: The K-theory of fields in characteristic p. Invent. Math. 139(3), 459–493 (2000)MATHMathSciNetGoogle Scholar
  33. 33.
    Geisser, T., Levine, M.: The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky. J. Reine Angew. Math. 530, 55–103 (2001)MATHMathSciNetGoogle Scholar
  34. 34.
    Goodwillie, T.G.: Relative algebraic K-theory and cyclic homology. Ann. Math. (2) 124(2), 347–402 (1986)MathSciNetGoogle Scholar
  35. 35.
    Grayson, D.: Higher algebraic K-theory. II (after Daniel Quillen). In: Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976). Lecture Notes in Mathematics, vol. 551, pp. 217–240. Springer, Berlin (1976)Google Scholar
  36. 36.
    Grayson, D.R.: Localization for flat modules in algebraic K-theory. J. Algebra 61(2), 463–496 (1979)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Grothendieck, A.: Éléments de géométrie algébrique. I. Le langage des schémas. Inst. Hautes Études Sci. Publ. Math. 4, 228 (1960)Google Scholar
  38. 38.
    Grothendieck, A.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ. Math. 11, 167 (1961)Google Scholar
  39. 39.
    Haesemeyer, C.: Descent properties of homotopy K-theory. Duke Math. J. 125(3), 589–620 (2004)MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Happel, D.: On the derived category of a finite-dimensional algebra. Comment. Math. Helv. 62(3), 339–389 (1987)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)Google Scholar
  42. 42.
    Hsiang, W.C.: Geometric applications of algebraic K-theory. In: Proceedings of the International Congress of Mathematicians, (Warsaw, 1983), vol.1,2, pp. 99–118. PWN, Warsaw (1984)Google Scholar
  43. 43.
    Hovey, M., Shipley, B., Smith, J.: Symmetric spectra. J. Am. Math. Soc. 13(1), 149–208 (2000)MATHMathSciNetGoogle Scholar
  44. 44.
    Kapranov, M.M.: On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92(3), 479–508 (1988)MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Karoubi, M.: Foncteurs dérivés et K-théorie. Catégories filtrées. C. R. Acad. Sci. Paris Sér. A-B 267, A328–A331 (1968)MathSciNetGoogle Scholar
  46. 46.
    Keller, B.: Chain complexes and stable categories. Manuscripta Math. 67(4), 379–417 (1990)MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Keller, B.: Derived categories and universal problems. Commun. Algebra 19(3), 699–747 (1991)MATHCrossRefGoogle Scholar
  48. 48.
    Keller, B.: Derived categories and their uses. In: Handbook of algebra, vol.1, pp. 671–701. North-Holland, Amsterdam (1996)Google Scholar
  49. 49.
    Keller, B.: On the cyclic homology of exact categories. J. Pure Appl. Algebra 136(1), 1–56 (1999)MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Keller, B.: On differential graded categories. In: International Congress of Mathematicians, vol. II, pp. 151–190. European Mathematical Society, Zürich (2006)Google Scholar
  51. 51.
    Kurihara, M.: Some remarks on conjectures about cyclotomic fields and K-groups of Z. Compos. Math. 81(2), 223–236 (1992)MATHMathSciNetGoogle Scholar
  52. 52.
    Kuznetsov, A.: Derived categories of quadric fibrations and intersections of quadrics. Adv. Math. 218(5), 1340–1369 (2008)MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Levine, M.: Bloch’s higher Chow groups revisited. In: K-theory (Strasbourg, 1992). Astérisque 226, 10, 235–320 (1994)Google Scholar
  54. 54.
    Levine, M.: K-theory and motivic cohomology of schemes. K-theory archive preprint 336 (1999)Google Scholar
  55. 55.
    Levine, M.: The homotopy coniveau tower. J. Topol. 1(1), 217–267 (2008)MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Lück, W., Reich, H.: The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory. In: Handbook of K-theory, vol. 1,2, pp. 703–842. Springer, Berlin (2005)CrossRefGoogle Scholar
  57. 57.
    Matsumura, H.: Commutative ring theory. In: Cambridge Studies in Advanced Mathematics, second edn., vol.8. Cambridge University Press, Cambridge (1989) Translated from the Japanese by M. Reid.Google Scholar
  58. 58.
    Peter May, J.: Simplicial objects in algebraic topology. In: Van Nostrand Mathematical Studies, vol. 11. D. Van Nostrand Co., Inc., Princeton (1967)Google Scholar
  59. 59.
    Milnor, J.: Introduction to algebraic K-theory. In: Annals of Mathematics Studies, vol.72. Princeton University Press, Princeton (1971)Google Scholar
  60. 60.
    Mitchell, S.A.: Hypercohomology spectra and Thomason’s descent theorem. In: Algebraic K-theory (Toronto, ON, 1996). Fields Institute Communications, vol.16, pp.221–277. American Mathematical Society, Providence, RI (1997)Google Scholar
  61. 61.
    MacLane, S.: Categories for the working mathematician. In: Graduate Texts in Mathematics, second edn., vol.5. Springer, New York (1998)Google Scholar
  62. 62.
    Mazza, C., Voevodsky, V., Weibel, C.: Lecture notes on motivic cohomology. Clay Mathematics Monographs, vol.2. American Mathematical Society, Providence, RI (2006)MATHGoogle Scholar
  63. 63.
    Neeman, A.: Some new axioms for triangulated categories. J. Algebra 139(1), 221–255 (1991)MATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Neeman, A.: The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. École Norm. Sup. (4) 25(5), 547–566 (1992)Google Scholar
  65. 65.
    Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Am. Math. Soc. 9(1), 205–236 (1996)MATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    Neeman, A.: Triangulated categories. In: Annals of Mathematics Studies, vol. 148. Princeton University Press, Princeton, NJ (2001)Google Scholar
  67. 67.
    Neeman, A., Ranicki, A.: Noncommutative localisation in algebraic K-theory. I. Geom. Topol. 8, 1385–1425 (electronic) (2004)Google Scholar
  68. 68.
    Nesterenko, Y.P., Suslin, A.A.: Homology of the general linear group over a local ring, and Milnor’s K-theory. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 121–146 (1989)MATHMathSciNetGoogle Scholar
  69. 69.
    Panin, I.A.: The equicharacteristic case of the Gersten conjecture. Tr. Mat. Inst. Steklova, 241(Teor. Chisel, Algebra i Algebr. Geom.) pp. 169–178 (2003)Google Scholar
  70. 70.
    Popescu, N.: Abelian categories with applications to rings and modules. London Mathematical Society Monographs, vol. 3. Academic Press, London (1973)Google Scholar
  71. 71.
    Pedrini, C., Weibel, C.: The higher K-theory of complex varieties. K-Theory. 21(4), 367–385 (2000) Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part VGoogle Scholar
  72. 72.
    Quillen, D.: On the cohomology and K-theory of the general linear groups over a finite field. Ann. Math. (2) 96, 552–586 (1972)Google Scholar
  73. 73.
    Quillen, D.: Higher algebraic K-theory. I. In: Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math., vol. 341, pp. 85–147. Springer, Berlin (1973)Google Scholar
  74. 74.
    Ranicki, A.: Noncommutative localization in topology. In: Non-commutative localization in algebra and topology. London Mathematics Society Lecture Note Series, vol. 330, pp. 81–102. Cambridge University Press, Cambridge (2006)Google Scholar
  75. 75.
    Reid, L.: N-dimensional rings with an isolated singular point having nonzero K N. K-Theory. 1(2), 197–205 (1987)Google Scholar
  76. 76.
    Samokhin, A.: Some remarks on the derived categories of coherent sheaves on homogeneous spaces. J. Lond. Math. Soc. (2) 76(1), 122–134 (2007)Google Scholar
  77. 77.
    Schlichting, M.: Hermitian K-theory, derived equivalences and Karoubi’s fundamental theorem. In preparationGoogle Scholar
  78. 78.
    Schlichting, M.: Witt groups of singular varieties. In preparation.Google Scholar
  79. 79.
    Schwede, S.: Book project about symmetric spectra.
  80. 80.
    Schlichting, M.: A note on K-theory and triangulated categories. Invent. Math. 150(1), 111–116 (2002)MATHCrossRefMathSciNetGoogle Scholar
  81. 81.
    Schlichting, M.: Delooping the K-theory of exact categories. Topology 43(5), 1089–1103 (2004)MATHCrossRefMathSciNetGoogle Scholar
  82. 82.
    Schlichting, M.: Negative K-theory of derived categories. Math. Z. 253(1), 97–134 (2006)MATHCrossRefMathSciNetGoogle Scholar
  83. 83.
    Schlichting, M.: The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes. Invent. Math. 179(2), 349–433 (2010)MATHCrossRefMathSciNetGoogle Scholar
  84. 84.
    Serre, J.-P.: Modules projectifs et espaces fibrés à fibre vectorielle. In: Séminaire, P., Dubreil, M.-L., Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23, p.18. Secrétariat mathématique, Paris (1958)Google Scholar
  85. 85.
    Théorie des intersections et théorème de Riemann-Roch. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6). Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de Ferrand, D., Jouanolou, J.P., Jussila, O., Kleiman, S., Raynaud, M. et Serre, J.P. Lecture Notes in Mathematics, vol. 225. Springer, Berlin (1971)Google Scholar
  86. 86.
    Sherman, C.: Group representations and algebraic K-theory. In: Algebraic K-theory, Part I (Oberwolfach, 1980). Lecture Notes in Mathematics, vol. 966, pp. 208–243. Springer, Berlin (1982)Google Scholar
  87. 87.
    Spaltenstein, N.: Resolutions of unbounded complexes. Compos. Math. 65(2), 121–154 (1988)MATHMathSciNetGoogle Scholar
  88. 88.
    Suslin, A., Voevodsky, V.: Bloch-Kato conjecture and motivic cohomology with finite coefficients. In: The arithmetic and geometry of algebraic cycles (Banff, AB, 1998). NATO Science Series C: Mathematical and Physical Sciences, vol. 548, pp. 117–189. Kluwer Academic, Dordrecht (2000)Google Scholar
  89. 89.
    Swan, R.G.: K-theory of quadric hypersurfaces. Ann. Math. (2), 122(1), 113–153 (1985)Google Scholar
  90. 90.
    Thomason, R.W.: Les K-groupes d’un schéma éclaté et une formule d’intersection excédentaire. Invent. Math. 112(1):195–215 (1993)MATHCrossRefMathSciNetGoogle Scholar
  91. 91.
    Thomason, R.W.: The classification of triangulated subcategories. Compos. Math. 105(1), 1–27 (1997)MATHCrossRefMathSciNetGoogle Scholar
  92. 92.
    Toen, B.: Lectures on DG-categories. In this volumeGoogle Scholar
  93. 93.
    Totaro, B.: Milnor K-theory is the simplest part of algebraic K-theory. K-Theory 6(2), 177–189 (1992)Google Scholar
  94. 94.
    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, MA (1990)Google Scholar
  95. 95.
    Verdier, J.-L.: Des catégories dérivées des catégories abéliennes. Astérisque 239, pp. xii+253 (1996) With a preface by Illusie, L., edited and with a note by Maltsiniotis,GGoogle Scholar
  96. 96.
    Voevodsky, V.: Motivic cohomology with Z∕2-coefficients. Publ. Math. Inst. Hautes Études Sci. (98), 59–104 (2003)Google Scholar
  97. 97.
    Vorst, T.: Localization of the K-theory of polynomial extensions. Math. Ann. 244(1), 33–53 (1979) With an appendix by Wilberd van der KallenGoogle Scholar
  98. 98.
    Voevodsky, V., Suslin, A., Friedlander, E.M.: In: Cycles, transfers, and motivic homology theories. Annals of Mathematics Studies, vol. 143. Princeton University Press, Princeton, NJ (2000)Google Scholar
  99. 99.
    Waldhausen, F.: Algebraic K-theory of generalized free products. I, II. Ann. Math. (2) 108(1), 135–204 (1978)Google Scholar
  100. 100.
    Waldhausen, F.: Algebraic K-theory of spaces. In: Algebraic and geometric topology (New Brunswick, N.J., 1983). Lecture Notes in Mathematics, vol. 1126, pp. 318–419. Springer, Berlin (1985)Google Scholar
  101. 101.
    Washington, L.C.: Introduction to cyclotomic fields. Graduate Texts in Mathematics, vol.83. Springer, New York (1982)Google Scholar
  102. 102.
    Weibel, C.A.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, vol.38. Cambridge University Press, Cambridge (1994)Google Scholar
  103. 103.
    Weibel, C.: Algebraic K-theory of rings of integers in local and global fields. In: Handbook of K-theory, vol. 1,2, pp. 139–190. Springer, Berlin (2005)Google Scholar
  104. 104.
    Whitehead, G.W.: In: Elements of homotopy theory. Graduate Texts in Mathematics, vol.61. Springer, New York (1978)Google Scholar

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Authors and Affiliations

  1. 1.Mathematics Institute, Zeeman BuildingUniversity of WarwickCoventryUK

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