Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T01:29:24.607Z Has data issue: false hasContentIssue false

5 - The Norm Residue Theorem and the Quillen-Lichtenbaum Conjecture

Published online by Cambridge University Press:  05 March 2015

By
Manfred Kolster
Edited by
John Coates ,
A. Raghuram ,
Anupam Saikia  and
R. Sujatha
Manfred Kolster
Affiliation:
McMaster University, Hamilton, Canada
John Coates
Affiliation:
University of Cambridge
A. Raghuram
Affiliation:
Indian Institute of Science Education and Research, Pune
Anupam Saikia
Affiliation:
Indian Institute of Technology, Guwahati
R. Sujatha
Affiliation:
University of British Columbia, Vancouver
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
The Bloch–Kato Conjecture for the Riemann Zeta Function , pp. 97 - 120
Publisher: Cambridge University Press
Print publication year: 2015

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[Be87] Beilinson, A. A. 1987. Height pairing between algebraic cycles. K-theory, arithmetic and geometry (Moscow, 1984–1986). Lecture Notes in Math., 1289, 1–25. Berlin: Springer.
[B180] Bloch, S. 1980. Lectures on Algebraic Cycles. Duke University Mathematics Series IV. Duke University.
[Bl86] Bloch, S. 1986. Algebraic cycles and higher K-theory. Adv. Math., 61, 267–304.Google Scholar
[Bl10] Bloch, S. 2010. Lectures on Algebraic Cycles, Second Edition. New Mathematical Monographs, 16. Cambridge, UK: Cambridge University Press.
[BL95] Bloch, S., and Lichtenbaum, S. 1995. A spectral sequence for motivic cohomology. Unpublished.Google Scholar
[BK90] Bloch, S., and Kato, K. 1986. p-adic cohomology. Publ.Math.de l'l.H.É.S., 63, 107–152.Google Scholar
[B077] Borel, A. 1977. Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup., 7, 613–636.Google Scholar
[BT73] Bass, H., and Tate, J. 1973. The Milnor ring of a global field (with an appendix on euclidean quadratic imaginary fields, by J. Tate). Alge-braic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math., 342, 349–446. Berlin: Springer.
[DF85] Dwyer, W, and Friedlander, E. 1985. Algebraic and etale K-theory. Trans. Amer. Math. Soc., 292 (1), 247–280.Google Scholar
[FS02] Friedlander, E., and Suslin, A. 2002. The spectral sequence relating algebraic K-theory to motivic cohomology. Ann. Sci. École Norm. Sup. (4), 35 (6), 773–875.Google Scholar
[Ge04] Geisser, Th. 2004. Motivic cohomology over Dedekind rings. Math. Z., 248 (4), 773–794.
[Ge05] Geisser, Th. 2005. Motivic Cohomology, K-Theory and Topological Cyclic Homology. Handbook of K-Theory, 193–234. Berlin, New York: Springer.
[GL00] Geisser, Th., and Levine, M. 2000. The p-part of K-theory of fields in characteristicp. Invent. Math., 139, 459–494.Google Scholar
[GL01] Geisser, Th., and Levine, M. 2001. The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky. J. Reine Angew. Math., 530, 55–103.Google Scholar
[GS06] Gille, Ph., and Szamuely, T. 2006. Central Simple Algebras and Galois Cohomolog. Cambridge Studies in Advanced Mathematics, 101. Cambridge UK: Cambridge University Press.
[GS99] Gillet, H., and SoulÉ, C. 1999. Filtrations on higher algebraic K-theory. Algebraic K-theory (Seattle, WA, 1997). Proc. Sympos. Pure Math., 67, 89–148. Providence, RI: Amer. Math. Soc.
[Gr95] Grayson, D. 1995. Weight filtrations via commuting automorphisms. K-Theory, 9 (2), 139–172.Google Scholar
[Gr05] Grayson, D. 2005. The Motivic Spectral Sequence. Handbook of K-Theory, 39–70. Berlin, New York: Springer.
[HW12] Haesemeyer, C., and Weibel, Ch.The Norm Residue Theorem in Motivic Cohomology. Preprint 2012.
[HW09] Haesemeyer, C., and Weibel, Ch. 2009. Norm varieties and the chain lemma (after Markus Rost). Abel Symposium, 95–130. Berlin: Springer.Google Scholar
[Ka97] Kahn, B. 1997. The Quillen-Lichtenbaum Conjecture at the prime 2. Unpublished.Google Scholar
[Ka05] Kahn, B. 2005. Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry. Handbook of K-Theory, 351–428. Berlin, New York: Springer.
[Kt80] Kato, K. 1980. A generalization of local class field theory by using K-groups. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (3), 603–683.Google Scholar
[Ko04] Kolster, M. 2004. K-theory and arithmetic. Pages 191–258 (electronic) of: Contemporary developments in algebraic K-theory. ICTP Lect. Notes, XV, Trieste.
[Le99] Levine, M. 1999. K-theory and motivic cohomology of schemes. Unpublished.Google Scholar
[Le01] Levine, M. 2001. Techniques of localization in the theory of algebraic cycles. J. Alg. Geom, 10, 299–363.Google Scholar
[Li73] Lichtenbaum, S. 1973. Values of zeta-functions, étale cohomology, and algebraic K-theory. Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972).Lecture Notes in Math., 342, 489–501. Berlin: Springer.
[Li83] Lichtenbaum, S. 1983. Values of zeta-functions at negative integers. Number Theory. Lecture Notes in Math., 1068, 127–138. Berlin: Springer.
[MVW06] Mazza, C., Voevodsky, V., and Weibel, Ch. 2006. Lecture notes on motivic cohomology. Clay Mathematics Monographs, 2. Providence, RI: American Mathematical Society.
[Me82] Merkurjev, A. S. 1982. On the norm residue symbol of degree 2. Soviet Math. Dokl., 1546–1551.Google Scholar
[MS83] Merkurjev, A. S., and Suslin, A. A. 1983. K-cohomology of Severi-Brauer varieties and the norm residue homomorphism (in Russian). English trans.: Math. USSR Izv., 21, 307–340.
[MS91] Merkurjev, A. S., and Suslin, A. A. 1991. The norm residue homomorphism of degree 3 (in Russian). English trans.: Math. USSR Izv., 36, 349–367.Google Scholar
[Mi70] Milnor, J. 1970. Algebraic K-theory and quadratic forms. Invent. math., 9, 318–344.Google Scholar
[Mi71] Milnor, J. 1971. Introduction to algebraic K-theory. Princeton, NJ: Princeton University Press. Annals of Mathematics Studies, no. 72.
[NS90] Nesterenko, Yu., and Suslin, A. A. 1990. Homology of the general linear group over a local ring, and Milnor's K-theory (in Russian). English trans.: Math. USSR Izv., 34, 121–146.Google Scholar
[OVV07] Orlov, D., Vishik, A., and Voevodsky, V. 2007. An exact sequence for KM/2 with applications to quadratic forms. Ann. of Math., 165 (1), 1–13.Google Scholar
[Q73] Quillen, D. 1973. Higher algebraic K-theory I. Algebraic K-theory, I: Higher K-Theories, (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math., 341, 81–147. Berlin: Springer.
[Q74] Quillen, D. 1974. Higher Algebraic K-theory. Proc. of the International Congress of Mathematicians, 1, 171–176. Vancouver, BC.
[RW00] Rognes, J., and Weibel, C. 2000. Two-primary algebraic K-theory of rings of integers in number fields. J. Amer. Math. Soc., 13 (1), 1–54. Appendix A by M., Kolster.Google Scholar
[Ro98] Rost, M. 1998. Chain lemma for splitting fields of symbols. Unpublished.Google Scholar
[Se79] Serre, J.-P. 1979. Local fields. Graduate Texts in Mathematics, 67. New York: Springer.
[So79] Soulé, C. 1979. K-theorie des anneaux d'entiers de corps de nombres et cohomologie etale. Invent. Math., 55 (3), 251–295.Google Scholar
[So95] Soulé, C. Feb. 8, 1995. Letter to Bloch and Lichtenbaum.
[Sr91] Srinivas, V. 1991. Algebraic K-Theory. Progress in Mathematics, 90. Boston: Birkhauser.
[SV00] Suslin, A., and Voevodsky, V. 2000. 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, 117–189. Dordrecht: Kluwer Acad. Publ.Google Scholar
[Su85] Suslin, A. A. 1985. Algebraic K-theory and the norm residue homomorphism. J. of Math. Sciences, 30 (6), 2556–2611.Google Scholar
[SJ06] Suslin, A. A., and Joukhovitski, S. 2006. Norm varieties. J. Pure Appl. Algebra, 206 (1–2), 245–276.Google Scholar
[Sw85] Swan, R. 1985. K-Theory of Quadric Hypersurfaces. Ann. of Math., 122 (1), 113–153.Google Scholar
[Ta76] Tate, John. 1976. Relations between K2 and Galois cohomology. Invent. Math., 36, 257–274.Google Scholar
[To92] Totaro, B. 1992. Milnor's K-theory is the simplest part of algebraic K-theory. K-theory, 6, 177–189.Google Scholar
[Vo96] Voevodsky, V. 1996. The Milnor Conjecture. Unpublished.Google Scholar
[Vo00a] Voevodsky, V. 2000a. Cohomological theory of presheaves with transfers. Cycles, transfers and motivic homology theories. Annals ofMathe-matics Studies, 143, 87–137. Princeton: Princeton University Press.
[Vo00b] Voevodsky, V. 2000b. Triangulated categories of motives over a field. Cycles, transfers, and motivic homology theories. Annals of Mathematical Studies, 143, 87–137. Princeton: Princeton University Press.
[Vo02] Voevodsky, V. 2002. Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic. Int. Mat. Res. Notices, 7, 331–355.Google Scholar
[Vo03] Voevodsky, V. 2003. Motivic cohomology with Z/2-coefficients. Publ. Math. de l'I.H.É.S., 98, 59–104.Google Scholar
[Vo11] Voevodsky, V. 2011. On motivic cohomology with Z/l-coefficients. Ann. of Math., 174, 401–438.Google Scholar
[VSF00] Voevodsky, V., Suslin, A., and Friedlander, E. M. 2000. Cycles, transfers, and motivic homology theories. Annals of Mathematics Studies, 143. Princeton: Princeton University Press.Google Scholar
[We09] Weibel, Ch. 2009. The norm residue isomorphism theorem. J. Topology, 2, 346–372.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×