Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Two-primary algebraic $K$-theory of rings of integers in number fields
HTML articles powered by AMS MathViewer

by J. Rognes, C. Weibel and appendix by M. Kolster PDF
J. Amer. Math. Soc. 13 (2000), 1-54 Request permission

Abstract:

We relate the algebraic $K$-theory of the ring of integers in a number field $F$ to its étale cohomology. We also relate it to the zeta-function of $F$ when $F$ is totally real and Abelian. This establishes the $2$-primary part of the “Lichtenbaum conjectures.” To do this we compute the $2$-primary $K$-groups of $F$ and of its ring of integers, using recent results of Voevodsky and the Bloch–Lichtenbaum spectral sequence, modified for finite coefficients in an appendix. A second appendix, by M. Kolster, explains the connection to the zeta-function and Iwasawa theory.
References
  • John Coates, $p$-adic $L$-functions and Iwasawa’s theory, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 269–353. MR 0460282
  • Pierre Deligne and Kenneth A. Ribet, Values of abelian $L$-functions at negative integers over totally real fields, Invent. Math. 59 (1980), no. 3, 227–286. MR 579702, DOI 10.1007/BF01453237
  • Neal Koblitz (ed.), Number theory related to Fermat’s last theorem, Progress in Mathematics, vol. 26, Birkhäuser, Boston, Mass., 1982. MR 685284
  • Ralph Greenberg, On $p$-adic $L$-functions and cyclotomic fields, Nagoya Math. J. 56 (1975), 61–77. MR 360536, DOI 10.1017/S002776300001638X
  • Ralph Greenberg, On $p$-adic $L$-functions and cyclotomic fields. II, Nagoya Math. J. 67 (1977), 139–158. MR 444614, DOI 10.1017/S0027763000022583
  • Ralph Greenberg, On $p$-adic Artin $L$-functions, Nagoya Math. J. 89 (1983), 77–87. MR 692344, DOI 10.1017/S0027763000020250
  • Cornelius Greither, Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 449–499 (English, with English and French summaries). MR 1182638, DOI 10.5802/aif.1299
  • Kenkichi Iwasawa, Lectures on $p$-adic $L$-functions, Annals of Mathematics Studies, No. 74, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0360526, DOI 10.1515/9781400881703
  • Kenkichi Iwasawa, On $\textbf {Z}_{l}$-extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246–326. MR 349627, DOI 10.2307/1970784
  • Manfred Kolster, A relation between the $2$-primary parts of the main conjecture and the Birch-Tate-conjecture, Canad. Math. Bull. 32 (1989), no. 2, 248–251. MR 1006753, DOI 10.4153/CMB-1989-036-8
  • Manfred Kolster, Thong Nguyen Quang Do, and Vincent Fleckinger, Twisted $S$-units, $p$-adic class number formulas, and the Lichtenbaum conjectures, Duke Math. J. 84 (1996), no. 3, 679–717. MR 1408541, DOI 10.1215/S0012-7094-96-08421-5
  • Stephen Lichtenbaum, On the values of zeta and $L$-functions. I, Ann. of Math. (2) 96 (1972), 338–360. MR 360527, DOI 10.2307/1970792
  • B. Mazur and A. Wiles, Class fields of abelian extensions of $\textbf {Q}$, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, DOI 10.1007/BF01388599
  • Thống Nguyễn-Quang-Đỗ, Une étude cohomologique de la partie $2$-primaire de $K_2{\scr O}$, $K$-Theory 3 (1990), no. 6, 523–542 (French, with English summary). MR 1071894, DOI 10.1007/BF01054449
  • Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
  • A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493–540. MR 1053488, DOI 10.2307/1971468
  • Klaus Haberland, Galois cohomology of algebraic number fields, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. With two appendices by Helmut Koch and Thomas Zink. MR 519872
  • J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI 10.1016/0040-9383(66)90004-8
  • Spencer Bloch, Algebraic cycles and higher $K$-theory, Adv. in Math. 61 (1986), no. 3, 267–304. MR 852815, DOI 10.1016/0001-8708(86)90081-2
  • S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology, Invent. Math. (to appear).
  • Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272 (1975). MR 387496, DOI 10.24033/asens.1269
  • J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
  • William G. Dwyer and Eric M. Friedlander, Algebraic and etale $K$-theory, Trans. Amer. Math. Soc. 292 (1985), no. 1, 247–280. MR 805962, DOI 10.1090/S0002-9947-1985-0805962-2
  • E. Friedlander and V. Voevodsky, Bivariant cycle cohomology, UIUC K-theory preprint server, no. 75, 1995.
  • Ofer Gabber, $K$-theory of Henselian local rings and Henselian pairs, Algebraic $K$-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989) Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 59–70. MR 1156502, DOI 10.1090/conm/126/00509
  • Cornelius Greither, Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 449–499 (English, with English and French summaries). MR 1182638, DOI 10.5802/aif.1299
  • B. Harris and G. Segal, $K_{i}$ groups of rings of algebraic integers, Ann. of Math. (2) 101 (1975), 20–33. MR 387379, DOI 10.2307/1970984
  • Raymond T. Hoobler, When is $\textrm {Br}(X)=\textrm {Br}^{\prime } (X)$?, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, Springer, Berlin-New York, 1982, pp. 231–244. MR 657433
  • Uwe Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), no. 2, 207–245. MR 929536, DOI 10.1007/BF01456052
  • Bruno Kahn, Some conjectures on the algebraic $K$-theory of fields. I. $K$-theory with coefficients and étale $K$-theory, Algebraic $K$-theory: connections with geometry and topology (Lake Louise, AB, 1987) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 279, Kluwer Acad. Publ., Dordrecht, 1989, pp. 117–176. MR 1045848
  • —, The Quillen–Lichtenbaum Conjecture at the prime $2$, UIUC K-theory preprint server, no. 208, 1997.
  • Stephen Lichtenbaum, On the values of zeta and $L$-functions. I, Ann. of Math. (2) 96 (1972), 338–360. MR 360527, DOI 10.2307/1970792
  • Stephen Lichtenbaum, 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., Vol. 342, Springer, Berlin, 1973, pp. 489–501. MR 0406981
  • James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
  • J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. MR 881804
  • Yu. P. Nesterenko and A. A. Suslin, Homology of the general linear group over a local ring, and Milnor’s $K$-theory, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 121–146 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 121–145. MR 992981, DOI 10.1070/IM1990v034n01ABEH000610
  • Jürgen Neukirch, Class field theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 280, Springer-Verlag, Berlin, 1986. MR 819231, DOI 10.1007/978-3-642-82465-4
  • 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, 878 (Russian). MR 864175
  • Daniel Quillen, On the cohomology and $K$-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972), 552–586. MR 315016, DOI 10.2307/1970825
  • Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
  • Daniel Quillen, Finite generation of the groups $K_{i}$ of rings of algebraic integers, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 179–198. MR 0349812
  • Daniel Quillen, Higher algebraic $K$-theory, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 171–176. MR 0422392
  • D. Quillen, Letter from Quillen to Milnor on $\textrm {Im}(\pi _{i}O\rightarrow \pi _{i}^{\textrm {s}}\rightarrow K_{i}\textbf {Z})$, Algebraic $K$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976) Lecture Notes in Math., Vol. 551, Springer, Berlin, 1976, pp. 182–188. MR 0482758
  • J. Rognes, Algebraic $K$-theory of the two-adic integers, J. Pure Appl. Algebra 134 (1999), 219–286.
  • J. Rognes and P. A. Østvær, Two-primary algebraic $K$-theory of two-regular number fields, Math. Z. (to appear).
  • J. Rognes and C. Weibel, Étale descent for two-primary algebraic $K$-theory of totally imaginary number fields, $K$-Theory 16 (1999), 101–104.
  • Peter Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), no. 2, 181–205 (German). MR 544704, DOI 10.1007/BF01214195
  • Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237, DOI 10.1007/978-1-4757-5673-9
  • C. Soulé, $K$-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1979), no. 3, 251–295 (French). MR 553999, DOI 10.1007/BF01406843
  • A. Suslin, On the $K$-theory of algebraically closed fields, Invent. Math. 73 (1983), no. 2, 241–245. MR 714090, DOI 10.1007/BF01394024
  • —, Higher Chow groups and étale cohomology, Preprint, 1994.
  • Andrei Suslin, Algebraic $K$-theory and motivic cohomology, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 342–351. MR 1403935
  • A. A. Suslin and V. Voevodsky, The Bloch–Kato conjecture and motivic cohomology with finite coefficients, UIUC K-theory preprint server, no. 83, 1995.
  • John Tate, Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 288–295. MR 0175892
  • V. Voevodsky, Triangulated categories of motives over a field, UIUC K-theory preprint server, no. 74, 1995.
  • —, The Milnor Conjecture, UIUC K-theory preprint server, no. 170, 1996.
  • J. B. Wagoner, Continuous cohomology and $p$-adic $K$-theory, Algebraic $K$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976) Lecture Notes in Math., Vol. 551, Springer, Berlin, 1976, pp. 241–248. MR 0498502
  • Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
  • Charles Weibel, Étale Chern classes at the prime $2$, Algebraic $K$-theory and algebraic topology (Lake Louise, AB, 1991) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 249–286. MR 1367303, DOI 10.1007/978-94-017-0695-7_{1}4
  • Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
  • Charles Weibel, The 2-torsion in the $K$-theory of the integers, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 6, 615–620 (English, with English and French summaries). MR 1447030, DOI 10.1016/S0764-4442(97)86977-7
  • A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493–540. MR 1053488, DOI 10.2307/1971468
Similar Articles
Additional Information
  • J. Rognes
  • Affiliation: Department of Mathematics, University of Oslo, Oslo, Norway
  • Email: rognes@math.uio.no
  • C. Weibel
  • Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903-2101
  • MR Author ID: 181325
  • Email: weibel@math.rutgers.edu
  • appendix by M. Kolster
  • Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
  • Email: kolster@mcmail.CIS.McMaster.CA
  • Received by editor(s): July 13, 1998
  • Published electronically: August 23, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 13 (2000), 1-54
  • MSC (2000): Primary 19D50; Secondary 11R70, 11S70, 14F20, 19F27
  • DOI: https://doi.org/10.1090/S0894-0347-99-00317-3
  • MathSciNet review: 1697095