Skip to main content
SpringerLink
Log in

The Completely Decomposed Topology on Schemes and Associated Descent Spectral Sequences in Algebraic K-Theory

  • Chapter
Book cover Algebraic K-Theory: Connections with Geometry and Topology

Part of the book series: NATO ASI Series ((ASIC,volume 279))

Abstract

Let X be a noetherian scheme of finite Krull dimension. A new Grothendieck topology on X, called the completely decomposed topology, is introduced, and the formalism of the corresponding cohomology and homotopy theories is developed. This formalism is applied to construct certain descent (or local-to-global) spectral sequences convergent to various algebraic K-groups of X, or to the homotopy groups of more general spectra. They refine the well-known Brown-Gersten spectral sequences.

To Alexander Grothendieck on his 60th birthday.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 379.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Grothendieck, A. and Dieudonne, J.: ‘Elements de Geometrie Algebrique’, IV, Publ. Math. I.H.E.S., 20, 24, 28, 32, 1964–66.

    Google Scholar 

  2. Seminaire de Geometrie Algebrique du Bois Marie:

    Google Scholar 

  3. Grothendieck, A.: ‘Revetements Etale et Groupes Fondamentales’, 1960–61, Lect. Notes in Mathematics 224, Berlin, Springer, 1971.

    Google Scholar 

  4. Grothendieck, A.: ‘Cohomologie locale des faisceaux coherents et Theoremes de Lefschetz locaux et globaux’, 1962, Amsterdam, North Holland, 1968.

    Google Scholar 

  5. Artin, M., Grothendieck, A. and Verdie, J.-L.: ‘Theorie des Topos et Cohomologie Etale des Schemas’, I-III, 1963–64, Lect.Notes in Math. 269, 270, 305, Berlin, Springer, 1972–1973.

    MATH  Google Scholar 

  6. Adams, J.F.: Stable homotopy and generalized homology, Chicago, Univ. of Chicago Press, 1974.

    Google Scholar 

  7. Artin, M. and Mazur, B.: ‘Etale Homotopy’, Lect. Notes in Math. 100, 2nd ed., Berlin, Springer, 1986.

    Google Scholar 

  8. Atiyah, M. and Hirzebruch, F.: ‘Vector bundles and homogeneous spaces’, Proc. of Symp. Pure Math. 3, “Differential Geometry”, Providence, A.M.S., 7–38, 1961.

    Google Scholar 

  9. Beilinsion, A.A.: ‘Higher regulators and values of L-functions’, in Itogi Nauki, Sovremennye Problemi Matematiki 24, Moskow, VINITI, 181–238, 1984.

    Google Scholar 

  10. Bousfield, A.K. and Kan, D.M.: ‘Homotopy Limits, Completions and Localizations’, Lect. Notes in Math. 304, Berlin, Springer, 1972.

    Book  MATH  Google Scholar 

  11. Breen, L.: ‘Extensions du Groupe additif’, Publ. Math. I.H.E.S. 48, 39–125, 1979.

    MathSciNet  Google Scholar 

  12. Brown, K.: ‘Abstract homotopy theory and generalized sheaf cohomology’, Trans. A.M.S. 186, 419–458, 1973.

    Article  Google Scholar 

  13. Brown, K. and Gersten, S.: ‘Algebraic K-theory as generalized sheaf cohomology’, Lect. Notes in Math. 341, Berlin, Springer, 266–292, 1973.

    Google Scholar 

  14. Deligne, P.: ‘Theorie de Hodge’, III, Publ. Math. I.H.E.S. 44, pp. 5–74, 1973.

    Google Scholar 

  15. Friedlander, E.: ‘Etale K-theory I. Connections with Etale Cohomology and Algebraic Vector Bundles’, Invent. Math. 60, 105–134, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  16. Friedlander, E.: ‘Etale K-theory II, Connections with Algebraic K-theory’, Ann. Sci. Ecole Norm. Sup. 15, 231–256, 1982.

    MathSciNet  MATH  Google Scholar 

  17. Gabber, O.: ‘K-theory of Henselian Local Rings and Henselian Pairs’, preprint, I.H.E.S. 1983.

    Google Scholar 

  18. Gersten, S.: ‘Problems about Higher K-functors’, in “Algebraic K-theory I,” Lecture Notes in Mathematics 341, Berlin, Springer, 43–57, 1973.

    Google Scholar 

  19. Gersten, S.: ‘Some exact sequences in the higher K-theory of rings’, in “Algebraic K-theory, I”, Lecture Notes in Math. 341, Berlin, Springer, 211–244, 1973.

    Google Scholar 

  20. Gersten, S.: ‘The Localization Theorem for Projective Modules’, Comm. in Algebra 2(4), 307–350, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  21. Gillet, H.: ‘Homological descent for the K-theory of coherent sheaves’, in Lecture Notes in Math. 1046, Berlin, Springer, 80–103, 1983.

    Google Scholar 

  22. Gillet, H.: ‘Gersten’s conjecture for the K-theory with torsion coefficients of a discrete valuation ring’, J. of Algebra 103, 377–380, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  23. Gillet, H. and Levine, M.: ‘The relative form of Gersten’s conjecture over a discrete valuation ring: the smooth case’, Journal of Pure and Appl. Algebra 46, 59–71, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  24. Gillet, H. and Soule, C.: ‘Filtrations on Higher Algebraic K-theory’, preprint 1984.

    Google Scholar 

  25. Gillet, H. and Thomason, R.: ‘The K-theory of Strict Hensel Local Rings and theorem of Suslin’, Journal Pure and Applied Algebra 34, 241–254, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  26. Godement, R.: Topologie Algebrique et Theorie des Faisceaux. Paris, Hermann, 1958.

    MATH  Google Scholar 

  27. Grayson, D. (after D. Quillen): ‘Higher Algebraic K-theory II’, Lect. notes in Math. 551, Berlin, Springer, 217–240, 1978.

    Book  Google Scholar 

  28. Grothendieck, A.: ‘Le groupe de Brauer, I-III’, in Pix exposes sur la cohomologie des schémas, Amsterdam, North Holland, 46–138, 1968.

    Google Scholar 

  29. Harstshorne, R.: ‘Residues and Duality’, Lect. Notes in Math., 20, Berlin Springer Verlag. 1966.

    Google Scholar 

  30. Kato, K. and Saito, S.: ‘Global class field theory of arithmetic schemes’, Contemporary Math. 55(I), 255–332, 1986.

    MathSciNet  Google Scholar 

  31. Lichtenbaum, S.: ‘Values of zeta functions, etale cohomology, and algebraic K-theory’, Lect. Notes in Math. 342, Berlin, Springer, 489–501, 1973.

    Google Scholar 

  32. Merkurjev, A. and Suslin, A.: ‘K-homology of Severi-Brauer varieties and the norm residue symbol’, Izvestia ANSSSR 46, No. 5, 1011–1046, 1984.

    Google Scholar 

  33. Nisnevich, Ye. A.: ‘Arithmetic and cohomological invariants of semisimple group schemes and compactifications of locally symmetric spaces’, Funct. Anal. and Appl. 14(1), 75–76, 1980.

    MathSciNet  Google Scholar 

  34. Nisnevich, Ye. A.: ‘On certain arithmetic and cohomological invariants of semisimple groups’, Preprint, Harvard Univ., 1982.

    Google Scholar 

  35. Nisnevich, Ye. A.: ‘Etale cohomology and arithmetic of semisimple groups’, Thesis, Harvard, Ch. I, II, 1982: Ch. I. Adeles and Grothendieck Topologies; Ch. II. Serre-Grothendieck conjecture on rationally trivial principal homogeneous spaces. (circulated separately as preprints)

    Google Scholar 

  36. Nisnevich, Ye. A.: ‘Espaces homogènes principaux rationnellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind’, Comp. Rend. Acad. Sci., Paris 299, No. 1, 5–8, 1984.

    MathSciNet  MATH  Google Scholar 

  37. Nisnevich, Ye. A.: ‘Espaces homogènes principaux rationnellement triviaux, pureté et arithmétique des schémes en groupes réductifs sur les extensions des anneaux de locaux régulier de dimension 2’, Comptes Rendus Acad. Sci.,Paris, 1988, to appear.

    Google Scholar 

  38. Quillen, D.: ‘Higher algebraic K-theory I’, Lect. Notes in Math. 341, Berlin, Springer, 85–147, 1973.

    Google Scholar 

  39. Quillen, D.: ‘Higher Algebraic K-theory’, Proc. Intern. Congress Math., Vancouver, 1, 171–176, 1974.

    Google Scholar 

  40. Suslin, A.A.: ‘On the K-theory of algebraically closed fields’, Invent. Math. 73, 241–248, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  41. Suslin, A.A.: ‘On the K-theory of local fields’, Journ. Pure and Appl. Algebra 34, 301–318, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  42. Switzer, R.M.: Algebraic Topology — Homotopy and Homology. Die Grundlehren der Math. Wissenschaften, b. 212, Berlin, Springer, 1975.

    MATH  Google Scholar 

  43. Thomason, R.W.: ‘Algebraic K-theory and etale cohomology’, Ann. Sci. E.N.S. 13, 437–552, 1986.

    Google Scholar 

  44. Thomason, R.W.: ‘Bott stability in algebraic K-theory’, in “Application of Algebraic K-theory to Algebraic Geometry and Number Theory”, Contemn. Math. 55(I), 389–406, 1986.

    Google Scholar 

  45. Beilinson, A.: ‘Height pairing between algebraic cycles’, in “Current Trends in Arithmetical Algebraic Geometry”, Contemp. Math. 67, 3–24, 1987.

    Google Scholar 

  46. Bousfield, A.K. and Friedlander, E.M.: ‘Homotopy theory of Γ-spaces, Spectra and Bisimplicial Sets’, Lecture Notes in Math. 658, Berlin, Springer, 80–130, 1978.

    Google Scholar 

  47. Browder, W.: ‘Algebraic K-theory with finite coefficients’, in ‘Geometric applications of Homotopy Theory I’, Lecture Notes in Math. 657, Berlin, Springer, 40–84, 1978.

    Chapter  Google Scholar 

  48. Harder, G.: ‘Halbeinfache Gruppenschemata liber Dedekindringen’, Invet. Math., 4, 165–191, 1967.

    Article  MathSciNet  MATH  Google Scholar 

  49. Harder, G.: ‘Eine Bemerkung zum schwachen approximationssatz’, Archiv der Mathematik, XIX, 465–471, 1968.

    Article  MathSciNet  Google Scholar 

  50. Illusie, L.: Complex contangent et deformations, I, II, Lect. Notes in Math. 239, 284, Berlin, Springer, 1972.

    Google Scholar 

  51. Jardine, J.F.: ‘Simplicial objects in a Grothendieck topos’, Contemp. Math. 55(I), 193–240, 1986.

    MathSciNet  Google Scholar 

  52. Kato, K.: letters to the author of March 11 and 12, 1987.

    Google Scholar 

  53. Lichtenbaum, S.: ‘Values of zeta-functions at non-negative integers’, Lect. Notes in Math. 1068, Berlin, Springer, 127–138, 1984.

    Google Scholar 

  54. Milne, J., Etale Cohomology, Princeton, Princeton Univ. Press, 1980.

    MATH  Google Scholar 

  55. Saito, S.: ‘Some observations on motivic cohomology of arithmetic schemes’, preprint, Univ. of California, Berkeley, 1987.

    Google Scholar 

  56. Saito, S.: ‘Arithmetic theory of arithmetic surfaces’, preprint, Univ. of California, Berkeley, 1987.

    Google Scholar 

  57. Segal, G.: ‘Classifying spaces and spectral sequences’, Publ. Math. I.H.E.S. 34, 105–112, 1968.

    MATH  Google Scholar 

  58. Seminaire C. Chevalley, Ecole Norm Super., 2, 1958, “Anneaux de Chow et applications”, Paris, Secretariat Math., 1958.

    Google Scholar 

  59. Serre, J.-P.: Cohomologie Galoisienne, 4th edit., Lect. Notes in Math. 5, Berlin, Springer, 1974.

    Google Scholar 

  60. Swan, R.: The Theory of Sheaves, Chicago Lectures in Mathematics, Chicago, Univ. of Chicago Press, 1964.

    MATH  Google Scholar 

  61. Bloch, S. and Ogus, A.: ‘Gersten’s conjecture and the homology of schemes’, Ann. Sci. Ecole Norm. Super., 4 e ser. 7, 181–202, 1974.

    MathSciNet  MATH  Google Scholar 

  62. Cartan, A. and Eilenberg, S.: Homological algebra, Princeton Univ. Press, Princeton, 1956.

    MATH  Google Scholar 

  63. Grothendieck, A.: ‘Sur Quelques points d’algebre homologique’, Tohoku Math. Journal 9, 119–221, 1957.

    MathSciNet  MATH  Google Scholar 

  64. Hilton, P. and Stammbach, U.: A course in Homo logical Algebra, Graduate Text in Math. 4, Berlin, Springer, 1971.

    Google Scholar 

  65. MacLane, S.: Categories for the Working Mathematician, Graduate Text in Math. 5, Berlin, Springer, 1971.

    Google Scholar 

  66. Quillen, D.: ‘Homotopical algebra’, Lect. Notes in Math. 43, Berlin, Springer, 1967.

    MATH  Google Scholar 

  67. Thomason, R. and Trobaugh, Th.: ‘Higher algebraic K-theory of schemes and derived categories’ (in preparation).

    Google Scholar 

  68. Colliot-Thélène, J.-L. and Sansuc, J.-J.: ‘Cohomologie des groupes de type multiplicatif sur les schemas réquliers’, Compt. Rend. Acad. Sci., Paris, 287, ser A., 449–452, 1978.

    MATH  Google Scholar 

  69. Dwyer, W. and Friedlander, E.M.: ‘Etale K-theory and arithmetic’, Trans. A.M.S. 292, 247–280, 1985.

    MathSciNet  MATH  Google Scholar 

  70. Hilton, P.: ‘Homotopy theory and duality’, New York, Gordon and Breach Science Publishers, 1965.

    Google Scholar 

  71. Jardine, J.F.: ‘Simplicial presheaves’, Journal Pure and Applied Algebra, 47, 35–87, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  72. Jardine, J.F.: ‘Stable homotopy theory of simplicial presheaves’, Canadian Journal of Math., 39, 733–747, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  73. Knezer, M.: ‘Schwache approximation in algebraichen Gruppen’, Colloque sur la theorie des Groupes Algebrique, Brussels, 41–52, 1962.

    Google Scholar 

  74. Knezer, M.: ‘Strong Approximation’, Proc. Symp. Pure Math, v. 9, A.M.S., Providence, 187–195, 1966.

    Google Scholar 

  75. May, P.: ‘E∞-spaces, Group Completions and Permutative Categories’, in “New Developments in Topology”, London Mathematical Society, Lecture Notes Series, Vol. 11, London, Cambridge Univ. Press, 1974.

    Google Scholar 

  76. Sansuc, J.-J.: ‘Groupe de Brauer et arithmetiques des groupes algebriques lineaires sur un corps de nombres’, Journ. fur der Reine und Angew. Math., 327, 12–80, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  77. Segal, G.: ‘Categories and Cohomology Theories’, Topology, 13, 293–312, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  78. Soule, C.: ‘K-theorie des anneaux d’entiers de corps de numbres et cohomologie etale’, Invent. Math., 55, 251–295, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  79. Snaith, V.P.: ‘Algebraic cobrodism and K-theory’, Memoirs A.M.S., No. 221, Providence, A.M.S., 1979.

    Google Scholar 

  80. Snaith, V.P.: ‘Unitary K-homo1ogy and Lichtenbaum-Qui11en conjecture on the Algebraic K-theory of schemes’, in ‘Algebraic Topology, Aahus, 1982’, Proceeding, Lect. Notes in Math., v. 1051, 128–155, Berlin, Springer, 1984.

    Chapter  Google Scholar 

  81. Snaith, V.P.: ‘Toward the Lichtenbaum-Quillen conjecture concerning the Algebraic K-theory of schemes’, in ‘Algebraic K-theory, Number Theory, and Analysis, Bielifeld, 1982, Proceedings’, Lect. Notes in Math., v. 1046, 349–356, Berlin, Springer, 1984.

    Chapter  Google Scholar 

  82. Suslin, A.A.: ‘Algebraic K-theory of fields’, Proc. Intern. Congress of Mathem., Berkeley, CA, 1986, I, Providence, A.M.S., 222–244, 1988.

    Google Scholar 

  83. Suslin, A.A.: ‘Algebraic K-theory and the Romomorphism of norm residue homomorphism’, Itogi Nauki Techniki, Sovr. Proble. Matem., 25, 115–209, 1984.

    MathSciNet  Google Scholar 

  84. Voskoresensky, V.E.: Algebraic Tori, Moskow, Nauka, 1977 (in Russian).

    Google Scholar 

  85. Wagoner, J.: ‘Delooping of classifying spaces in algebraic K-theory’, Topology, 11, 349–370, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  86. Fiedorowicz, Z.: ‘A note on the spectra of algebraic K-theory’, Topology, 16, 417–421, 1972.

    Article  MathSciNet  Google Scholar 

  87. Giraud, J.: Cohomologie non-abelienne, Die Grund lehren der Math. Wissench., 179, Berlin, Springer, 1971.

    MATH  Google Scholar 

  88. Johnstone, P.T.: Topos Theory, New York — London, Academic Press, 1977.

    MATH  Google Scholar 

  89. May, J.P.: Simplicial Objects in Algebraic Topology. Princeton, Van Nostrand, 1967.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The John Hopkins University, Baltimore, MD, 21218, USA

    Ye. A. Nisnevich

Authors
  1. Ye. A. Nisnevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematics Department, University of Western Ontario, London, Ontario, Canada

    J. F. Jardine

  2. Mathematics Department, McMaster University, Hamilton, Ontario, Canada

    V. P. Snaith

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Kluwer Academic Publishers

About this chapter

Cite this chapter

Nisnevich, Y.A. (1989). The Completely Decomposed Topology on Schemes and Associated Descent Spectral Sequences in Algebraic K-Theory. In: Jardine, J.F., Snaith, V.P. (eds) Algebraic K-Theory: Connections with Geometry and Topology. NATO ASI Series, vol 279. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2399-7_11

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-2399-7_11

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7580-0

  • Online ISBN: 978-94-009-2399-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation