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Handbook of K-Theory pp 193–234Cite as

Motivic Cohomology, K-Theory and Topological Cyclic Homology

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Abstract

We give a survey onmotivic cohomology, higher algebraic K-theory, and topological cyclic homology. We concentrate on results which are relevant for applications in arithmetic algebraic geometry (in particular, we do not discuss non-commutative rings), and our main focus lies on sheaf theoretic results for smooth schemes, which then lead to global results using local-to-global methods.

Keywords

  • Irreducible Component
  • Spectral Sequence
  • Homotopy Group
  • Discrete Valuation Ring
  • Minimal Prime Ideal

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References

  1. A.A. Beilinson, Letter to Soulé, 1982.

    Google Scholar 

  2. A.A. Beilinson, Height pairings between algebraic cycles. K-theory, arithmetic and geometry (Moscow, 1984–1986), 1–25, Lecture Notes in Math. 1289, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  3. P. Berthelot, A. Grothendieck et L. Illusie, Théorie des intersections et théorème de Riemann–Roch. Séminaire de Géometrie Algèbrique du Bois–Marie 1966–1967. Lecture Notes in Mathematics 225, Springer-Verlag, Berlin, 1971.

    Google Scholar 

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

    MATH  CrossRef  MathSciNet  Google Scholar 

  5. S. Bloch, Algebraic Cycles and Higher K-theory. Adv. in Math. 61 (1986), 267–304.

    MATH  CrossRef  MathSciNet  Google Scholar 

  6. S. Bloch, A note on Gersten’s conjecture in the mixed characteristic case. Appl. of Alg. K-theory to Alg. Geometry and Number Theory, Cont. Math. AMS 55 Part I (1986), 75–78.

    Google Scholar 

  7. S. Bloch, Algebraic cycles and the Beilinson conjectures. The Lefschetz centennial conference, Part I (Mexico City, 1984), Cont. Math. AMS, 58, (1986), 65–79.

    Google Scholar 

  8. S. Bloch, The moving lemma for higher Chow groups. J. Alg. Geom. 3 (1994), no. 3, 537–568.

    MATH  MathSciNet  Google Scholar 

  9. S. Bloch, K. Kato, p-adic étale cohomology. Inst. Hautes Etudes Sci. Publ. Math. 63 (1986), 147–164.

    CrossRef  MathSciNet  Google Scholar 

  10. S. Bloch, S. Lichtenbaum, A spectral sequence for motivic cohomology. Preprint 1995.

    Google Scholar 

  11. S. Bloch, A. Ogus, Gersten’s conjecture and the homology of schemes. Ann. Sci. Ecole Norm. Sup. (4) 7 (1974), 181–201.

    MATH  MathSciNet  Google Scholar 

  12. M. Bökstedt, W.C. Hsiang, I. Madsen, The cyclotomic trace and algebraic K-theory of spaces. Invent. Math. 111 (1993), 465–539.

    MATH  CrossRef  MathSciNet  Google Scholar 

  13. A. Borel, Stable real cohomology of arithmetic groups. Ann. Sci. Ecole Norm. Sup. (4) 7 (1974), 235–272.

    MATH  MathSciNet  Google Scholar 

  14. A.K. Bousfield, D.M. Kan, Homotopy limits, completions and localizations. Lecture Notes in Math. 304, Springer-Verlag, Berlin, 1972.

    MATH  Google Scholar 

  15. K.S. Brown, S.M. Gersten, Algebraic K-theory as generalized sheaf cohomology. Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 266–292. Lecture Notes in Math. 341, Springer-Verlag, Berlin, 1973.

    Google Scholar 

  16. G. Carlson, Deloopings in Algebraic K-theory. This volume.

    Google Scholar 

  17. J.L. Colliot-Thélène, R. Hoobler, B. Kahn, The Bloch–Ogus–Gabber theorem. Fields Inst. Com. 16 (1997), 31–94.

    Google Scholar 

  18. W.G. Dwyer, E.M. Friedlander, Algebraic and etale K-theory. Trans. Amer. Math. Soc. 292 (1985), 247–280.

    MATH  CrossRef  MathSciNet  Google Scholar 

  19. Ph. Elbaz-Vincent, S. Muller-Stach, Milnor K-theory of rings, higher Chow groups and applications. Invent. Math. 148 (2002), no. 1, 177–206.

    MATH  CrossRef  MathSciNet  Google Scholar 

  20. J.M. Fontaine, W. Messing, p-adic periods and p-adic étale cohomology. Contemp. Math. 67 (1987), 179–207.

    MathSciNet  Google Scholar 

  21. E. Friedlander, A. Suslin, The spectral sequence relating algebraic K-theory to motivic cohomology. Ann. Sci. Ecole Norm. Sup. (4) 35 (2002), no. 6, 773–875.

    MATH  MathSciNet  Google Scholar 

  22. Fujiwara, A proof of the absolute purity conjecture (after Gabber). Algebraic geometry 2000, Azumino, 153–183, Adv. Stud. Pure Math., 36, Math. Soc. Japan, Tokyo, 2002.

    Google Scholar 

  23. W. Fulton, Intersection Theory. Springer 1984.

    Google Scholar 

  24. O. Gabber, K-theory of Henselian local rings and Henselian pairs. Algebraic K-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), Cont. Math. AMS 126 (1992), 59–70.

    Google Scholar 

  25. T. Geisser, p-adic K-theory of Hecke characters of imaginary quadratic fields. Duke Math. J. 86 (1997), 197–238.

    MATH  CrossRef  MathSciNet  Google Scholar 

  26. T. Geisser, Tate’s conjecture, algebraic cycles and rational K-theory in characteristic p. K-theory 13 (1998), 109–122.

    MATH  CrossRef  MathSciNet  Google Scholar 

  27. T. Geisser, On K 3 of Witt vectors of length two over finite fields. K-Theory 12 (1997), no. 3, 193–226.

    MATH  CrossRef  MathSciNet  Google Scholar 

  28. T. Geisser, Motivic cohomology over Dedekind rings. Math. Z. 248 (2004), 773–794.

    MATH  CrossRef  MathSciNet  Google Scholar 

  29. T. Geisser, Weil-étale cohomology over finite fields. Math. Ann. 330 (2004), 665–691.

    MATH  CrossRef  MathSciNet  Google Scholar 

  30. T. Geisser, L. Hesselholt, Topological Cyclic Homology of Schemes. K-theory, Proc. Symp. Pure Math. AMS 67 (1999), 41–87.

    Google Scholar 

  31. T. Geisser, L. Hesselholt, K-theory and topological cyclic homology of smooth schemes over discrete valuation rings. To appear in: Trans. AMS.

    Google Scholar 

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

    Google Scholar 

  33. T. Geisser, M. Levine, The p-part of K-theory of fields in characteristic p. Inv. Math. 139 (2000), 459–494.

    MATH  CrossRef  MathSciNet  Google Scholar 

  34. T. Geisser, M. Levine, The Bloch–Kato conjecture and a theorem of Suslin–Voevodsky. J. Reine Angew. Math. 530 (2001), 55–103.

    MATH  MathSciNet  Google Scholar 

  35. H. Gillet, Gersten’s conjecture for the K-theory with torsion coefficients of a discrete valuations ring. J. Alg. 103 (1986), 377–380.

    MATH  CrossRef  MathSciNet  Google Scholar 

  36. H. Gillet, M. Levine, The relative form of Gersten’s conjecture over a discrete valuation ring: The smooth case. J. of Pure Appl. Alg. 46 (1987), 59–71.

    MATH  CrossRef  MathSciNet  Google Scholar 

  37. H. Gillet, C. Soule, Filtrations on higher algebraic K-theory. Algebraic K-theory (Seattle, WA, 1997), 89–148, Proc. Sympos. Pure Math., 67, Amer. Math. Soc., Providence, RI, 1999.

    Google Scholar 

  38. D. Grayson, Higher algebraic K-theory. II (after Daniel Quillen). Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), pp. 217–240. Lecture Notes in Math. 551, Springer-Verlag, Berlin, 1976.

    Google Scholar 

  39. D. Grayson, Weight filtrations via commuting automorphisms. K-theory 9 (1995), 139–172.

    MATH  CrossRef  MathSciNet  Google Scholar 

  40. M. Gros, Classes de Chern et classes de cycles en cohomologie de Hodge–Witt logarithmique. Mém. Soc. Math. France (N.S.) 21 (1985).

    Google Scholar 

  41. M. Gros, N. Suwa, La conjecture de Gersten pour les faisceaux de Hodge–Witt logarithmique. Duke Math. J. 57 (1988), 615–628.

    MATH  CrossRef  MathSciNet  Google Scholar 

  42. A. Grothendieck, Eléments de géométrie algébrique. IV. Etude locale des schémas et des morphismes de schémas IV. Inst. Hautes Etudes Sci. Publ. Math. No. 32 (1967).

    Google Scholar 

  43. L. Hesselholt, On the p-typical curves in Quillen’s K-theory. Acta Math. 177 (1996), 1–53.

    MATH  CrossRef  MathSciNet  Google Scholar 

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

    Google Scholar 

  45. L. Hesselholt, K-theory of truncated polynomial algebras. This volume.

    Google Scholar 

  46. L. Hesselholt, I. Madsen, On the K-theory of finite algebras over Witt vectors of perfect fields. Topology 36 (1997), no. 1, 29–101.

    MATH  CrossRef  MathSciNet  Google Scholar 

  47. L. Hesselholt, I. Madsen, Cyclic polytopes and the K-theory of truncated polynomial algebras. Invent. Math. 130 (1997), no. 1, 73–97.

    MATH  CrossRef  MathSciNet  Google Scholar 

  48. L. Hesselholt, I. Madsen, The K-theory of local fields. Annals of Math. 158 (2003), 1–113.

    MATH  CrossRef  MathSciNet  Google Scholar 

  49. M. Hovey, B. Shipley, J. Smith, Symmetric spectra. J. Amer. Math. Soc. 13 (2000), no. 1, 149–208.

    MATH  CrossRef  MathSciNet  Google Scholar 

  50. L. Illusie, Complexe de de Rham–Witt et cohomologie cristalline. Ann. Sci. Ecole Norm. Sup. 12 (1979), 501–661.

    MATH  MathSciNet  Google Scholar 

  51. U. Jannsen, Continuous étale cohomology. Math. Ann. 280 (1988), 207–245.

    MATH  CrossRef  MathSciNet  Google Scholar 

  52. U. Jannsen, Motives, numerical equivalence, and semi-simplicity. Invent. Math. 107 (1992), no. 3, 447–452.

    MATH  CrossRef  MathSciNet  Google Scholar 

  53. B. Kahn, The Geisser–Levine method revisited and algebraic cycles over a finite field. Math. Ann. 324 (2002), 581–617.

    MATH  CrossRef  MathSciNet  Google Scholar 

  54. B. Kahn, Algebraic K-theory, algebraic cycles and arithmetic geometry. This volume.

    Google Scholar 

  55. K. Kato, Galois cohomology of complete discrete valuation fields. Lecture Notes Math. 967 (1982), 215–238.

    CrossRef  Google Scholar 

  56. M. Kurihara, A Note on p-adic Etale Cohomology. Proc. Jap. Acad. 63A (1987), 275–278.

    CrossRef  MathSciNet  Google Scholar 

  57. M. Levine, Mixed motives. Mathematical Surveys and Monographs, 57. American Mathematical Society, Providence, RI, 1998.

    MATH  Google Scholar 

  58. M. Levine, Techniques of localization in the theory of algebraic cycles. J. Alg. Geom. 10 (2001), 299–363.

    MATH  Google Scholar 

  59. M. Levine, K-theory and Motivic Cohomology of Schemes I. Preprint 2002.

    Google Scholar 

  60. M. Levine, The homotopy coniveau filtration. Preprint 2003.

    Google Scholar 

  61. M. Levine, Mixed motives. This volume.

    Google Scholar 

  62. S. Lichtenbaum, Values of zeta-functions at non-negative integers. In: Number theory, Noordwijkerhout 1983, 127–138, Lecture Notes in Math. 1068, Springer-Verlag, Berlin, 1984.

    Google Scholar 

  63. S. Lichtenbaum, Motivic complexes. Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, (1994), 303–313.

    Google Scholar 

  64. S. Lichtenbaum, The Weil-étale topology. Preprint 2001.

    Google Scholar 

  65. J.L. Loday, Cyclic homology. Grundlehren der Mathematischen Wissenschaften 301. Springer-Verlag, Berlin, 1992.

    MATH  Google Scholar 

  66. H. Matsumura, Commutative Ring Theory. Cambridge Studies in Advanced Mathematics 8 (1986).

    Google Scholar 

  67. R. McCarthy, Relative algebraic K-theory and topological cyclic homology. Acta Math. 79 (1997), 197–222.

    CrossRef  MathSciNet  Google Scholar 

  68. J.S. Milne, Arithmetic duality theorems. Perspectives in Mathematics 1. Academic Press, Inc., Boston, MA, 1986.

    MATH  Google Scholar 

  69. J.S. Milne, Values of zeta functions of varieties over finite fields. Amer. J. Math. 108 (1986), no. 2, 297–360.

    MATH  CrossRef  MathSciNet  Google Scholar 

  70. J.S. Milne, Motivic cohomology and values of zeta functions. Comp. Math. 68 (1988), 59–102.

    MATH  MathSciNet  Google Scholar 

  71. J.S. Milne, Etale cohomology. Princeton Math. Series 33.

    Google Scholar 

  72. J.S. Milne, Motives over finite fields. Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Part 1, (1994), 401–459.

    Google Scholar 

  73. J. Milnor, Algebraic K-theory and quadratic forms. Invent. Math. 9 (1969/1970), 318–344.

    Google Scholar 

  74. Y. Nesterenko, A. Suslin, Homology of the general linear group over a local ring, and Milnor’s K-theory. Math. USSR-Izv. 34 (1990), no. 1, 121–145.

    MATH  CrossRef  MathSciNet  Google Scholar 

  75. Y. Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory. Algebraic K-theory: connections with geometry and topology (Lake Louise, AB, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 279 (1989), 241–342.

    Google Scholar 

  76. I.A. Panin, The Hurewicz theorem and K-theory of complete discrete valuation rings. Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 763–775.

    MATH  MathSciNet  Google Scholar 

  77. D. Popescu, General Neron Desingularisation and Approximation (Letter to the Editor). Nagoya Math. J. 118 (1990), 45–53.

    MATH  MathSciNet  Google Scholar 

  78. D. Quillen, On the cohomology and K-theory of the general linear groups over a finite field. Ann. of Math. (2) 96 (1972), 552–586.

    CrossRef  MathSciNet  Google Scholar 

  79. D. Quillen, Higher algebraic K-theory. In: Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), 85–147. Lecture Notes in Math. 341, Springer-Verlag, Berlin, 1973.

    Google Scholar 

  80. P. Roberts, Multiplicities and Chern classes in local algebra, Cambridge Tracts in Mathematics 133 (1998).

    Google Scholar 

  81. K. Sato, An étale Tate twist with finite coefficients and duality in mixed characteristic. Preprint 2002.

    Google Scholar 

  82. P. Schneider, p-adic points of motives. Motives (Seattle, WA, 1991), Proc. Symp. Pure Math. AMS 55, Part 2, Amer. Math. Soc., Providence, RI (1994), 225–249.

    Google Scholar 

  83. J.P. Serre, Algébre locale. Multiplicités. (French) Cours au Collège de France, 1957–1958, redige par Pierre Gabriel. Seconde edition, 1965. Lecture Notes in Mathematics 11, Springer-Verlag, Berlin, 1965.

    Google Scholar 

  84. C. Soulé, K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale. Inv. Math. 55 (1979), 251–295.

    MATH  CrossRef  Google Scholar 

  85. C. Soulé, Groupes de Chow et K-théorie de variétés sur un corps fini. Math. Ann. 268 (1984), 317–345.

    MATH  CrossRef  MathSciNet  Google Scholar 

  86. C. Soulé, Operations on etale K-theory. Applications. Algebraic K-theory, Part I (Oberwolfach, 1980), 271–303, Lecture Notes in Math. 966, Springer-Verlag, Berlin, 1982.

    Google Scholar 

  87. A. Suslin, Higher Chow groups and étale cohomology. Ann. Math. Studies 132 (2000), 239–554.

    MathSciNet  Google Scholar 

  88. A. Suslin, On the K-theory of local fields. J. Pure Appl. Alg. 34 (1984), 301–318.

    MATH  CrossRef  MathSciNet  Google Scholar 

  89. A. Suslin, Algebraic K-theory and motivic cohomology. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), 342–351, Birkhauser, Basel, 1995.

    Google Scholar 

  90. A. Suslin, On the Grayson spectral sequence. Preprint 2002.

    Google Scholar 

  91. A. Suslin, V. Voevodsky, Bloch–Kato conjecture and motivic cohomology with finite coefficients. The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., 548 (2000), 117–189.

    Google Scholar 

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

    Google Scholar 

  93. J. Tate, Relations between K 2 and Galois ohomology. Invent. Math. 36 (1976), 257–274.

    MATH  CrossRef  MathSciNet  Google Scholar 

  94. R.W. Thomason, Bott stability in algebraic K-theory. Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), 389–406, Cont. Math. AMS 55, (1986).

    Google Scholar 

  95. R.W. Thomason, Algebraic K-theory and etale cohomology. Ann. Sci. Ecole Norm. Sup. 18 (1985), 437–552.

    MATH  MathSciNet  Google Scholar 

  96. R.W. Thomason, Les K-groupes d’un schéma éclaté et une formule d’intersection excédentaire. Invent. Math. 112 (1993), no. 1, 195–215.

    MATH  CrossRef  MathSciNet  Google Scholar 

  97. R. Thomason, T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift, Vol. III, 247–435, Progr. Math., 88, Birkhäuser Boston, Boston, MA, 1990.

    Google Scholar 

  98. B. Totaro, Milnor K-theory is the simplest part of algebraic K-theory. K-Theory 6 (1992), no. 2, 177–189.

    MATH  CrossRef  MathSciNet  Google Scholar 

  99. V. Voevodsky, Homology of schemes. Selecta Math. (N.S.) 2 (1996), no. 1, 111–153.

    MATH  CrossRef  MathSciNet  Google Scholar 

  100. V. Voevodsky, Triangulated categories of motives over a field. Cycles, transfers, and motivic homology theories, 188–238, Ann. of Math. Stud., 143, Princeton Univ. Press, Princeton, NJ, 2000.

    Google Scholar 

  101. V. Voevodsky, On 2-torsion in motivic cohomology. Preprint 2001.

    Google Scholar 

  102. V. Voevodsky, Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic. Int. Math. Res. Not. 7 (2002), 351–355.

    CrossRef  MathSciNet  Google Scholar 

  103. V. Voevodsky, On motivic cohomology with Z/l-coefficients. Preprint 2003.

    Google Scholar 

  104. F. Waldhausen, Algebraic K-theory of generalized free products. I, II. Ann. of Math. (2) 108 (1978), no. 1, 135–204.

    CrossRef  MathSciNet  Google Scholar 

  105. F. Waldhausen, Algebraic K-theory of spaces. Algebraic and geometric topology (New Brunswick, N.J., 1983), 318–419, Lecture Notes in Math. 1126, Springer-Verlag, Berlin, 1985.

    Google Scholar 

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

    MATH  Google Scholar 

  107. C. Weibel, Products in higher Chow groups and motivic cohomology. Algebraic K-theory (Seattle, WA, 1997), 305–315, Proc. Sympos. Pure Math. 67, Amer. Math. Soc., Providence, RI, 1999.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, 90089-2532, Los Angeles, CA, USA

    Thomas H. Geisser

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  1. Thomas H. Geisser
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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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Geisser, T. (2005). Motivic Cohomology, K-Theory and Topological Cyclic Homology. In: Friedlander, E., Grayson, D. (eds) Handbook of K-Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27855-9_6

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