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Two-primary algebraic $K$-theory of rings of integers in number fields

Author(s): J. Rognes; C. Weibel; appendix by M. Kolster
Journal: J. Amer. Math. Soc. 13 (2000), 1-54.
MSC (2000): Primary 19D50; Secondary 11R70, 11S70, 14F20, 19F27
Posted: August 23, 1999
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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:

[A]
J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21-71. MR 33:6628

[Bl]
S. Bloch, Algebraic cycles and higher $K$-theory, Adv. Math. 61 (1986), 267-304. MR 88f:18010

[BL]
S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology, Invent. Math. (to appear).

[Bo]
A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. Ecole Norm. Sup. 7 (1974), 235-272. MR 52:8338

[CF]
J. W. S. Cassels and A. Frölich, eds., Algebraic Number Theory, Academic Press, 1967. MR 35:6500

[DF]
W. Dwyer and E. Friedlander, Algebraic and étale $K$-theory, Trans. AMS 292 (1985), 247-280. MR 87h:18013

[FV]
E. Friedlander and V. Voevodsky, Bivariant cycle cohomology, UIUC K-theory preprint server, no. 75, 1995.

[Ga]
O. Gabber, $K$-theory of Henselian local rings and Henselian pairs, AMS Contemp. Math., vol. 126, 1992, pp. 59-70. MR 93c:19005

[Gri]
C. Greither, Class groups of Abelian fields, and the Main Conjecture, Ann. Inst. Fourier, Grenoble 42 (1992), 449-499. MR 93j:11071

[HS]
B. Harris and G. Segal, $K_{i}$ groups of rings of algebraic integers, Annals of Math. 101 (1975), 20-33. MR 52:8222

[H]
R. T. Hoobler, When is $\operatorname {Br}(X) = \operatorname {Br}'(X)$ ?, Lecture Notes in Math., vol. 917, Springer Verlag, 1982, pp. 231-244. MR 83g:14006

[J]
U. Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), 207-245. MR 89a:14022

[K1]
B. Kahn, Some conjectures on the algebraic $K$-theory of fields I: $K$-theory with coefficients and étale cohomology, Algebraic $K$-theory: Connections with Geometry and Topology (J. F. Jardine and V. P. Snaith, eds.), NATO ASI Series C, vol. 279, Kluwer, 1989, pp. 117-176. MR 91d:19006

[K2]
-, The Quillen-Lichtenbaum Conjecture at the prime $2$, UIUC K-theory preprint server, no. 208, 1997.

[L1]
S. Lichtenbaum, On the values of zeta and $L$-functions, I, Annals of Math. 96 (1972), 338-360. MR 50:12975

[L2]
-, Values of zeta functions, étale cohomology, and algebraic $K$-theory, Lecture Notes in Math., vol. 342, Springer Verlag, 1973, pp. 489-501. MR 53:10765

[M1]
J. S. Milne, Étale Cohomology, Princeton University Press, 1980. MR 81j:14002

[M2]
-, Arithmetic Duality Theorems, Academic Press, 1986. MR 88e:14028

[NS]
Yu. P. Nesterenko and A. A. Suslin, Homology of the general linear group over a local ring, and Milnor's $K$-theory (Russian), Izv. Akad. Nauk. SSSR Ser. Mat. 53 (1989), 121-146. MR 90a:20092

[N]
J. Neukirch, Class Field Theory, Grundlehren der mathematischen Wissenschaften, vol. 280, Springer Verlag, 1986. MR 87i:11005

[P]
I. Panin, On a theorem of Hurewicz and $K$-theory of complete discrete valuation rings, Math. USSR Izvestiya 29 (1987), 119-131. MR 88a:18021

[Q1]
D. Quillen, On the cohomology and $K$-theory of the general linear group over a finite field, Annals of Math. 96 (1972), 552-586. MR 47:3565

[Q2]
-, Higher algebraic $K$-theory, I, Lecture Notes in Math., vol. 341, Springer Verlag, 1973, pp. 85-147. MR 49:2895

[Q3]
-, Finite generation of the groups $K_{i}$ of rings of algebraic integers, Lecture Notes in Math., vol. 341, Springer Verlag, 1973, pp. 179-198. MR 50:2305

[Q4]
-, Higher algebraic $K$-theory, Proc. Intern. Congress Math., Vancouver, 1974, vol. I, Canad. Math. Soc., 1975, pp. 171-176. MR 54:10382

[Q5]
-, Letter from Quillen to Milnor on $\operatorname {Im}(\pi _{i} O \to \pi _{i}^{S} \to K_{i}\mathbb Z)$, Lecture Notes in Math., vol. 551, Springer Verlag, 1976, pp. 182-188. MR 58:2811

[R]
J. Rognes, Algebraic $K$-theory of the two-adic integers, J. Pure Appl. Algebra 134 (1999), 219-286. CMP 99:06

[RO]
J. Rognes and P. A. Østvær, Two-primary algebraic $K$-theory of two-regular number fields, Math. Z. (to appear).

[RW]
J. Rognes and C. Weibel, Étale descent for two-primary algebraic $K$-theory of totally imaginary number fields, $K$-Theory 16 (1999), 101-104. CMP 99:08

[Sc]
P. Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), 181-205. MR 81i:12010

[Se]
J.-P. Serre, Local Fields, Graduate Texts in Mathematics, vol. 67, Springer Verlag, 1979. MR 82e:12016

[So]
C. Soulé, $K$-théorie des anneaux d'entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1979), 251-295. MR 81i:12016

[S1]
A. A. Suslin, On the $K$-theory of algebraically closed fields, Invent. Math. 73 (1983), 241-245. MR 85h:18008a

[S2]
-, Higher Chow groups and étale cohomology, Preprint, 1994.

[S3]
-, Algebraic $K$-theory and motivic cohomology, Proc. Intern. Congress Math., Zürich, 1994, Birkhäuser, 1995, pp. 342-351. MR 98g:19007

[SV]
A. A. Suslin and V. Voevodsky, The Bloch-Kato conjecture and motivic cohomology with finite coefficients, UIUC K-theory preprint server, no. 83, 1995.

[T]
J. Tate, Duality theorems in Galois cohomology over number fields, Proc. Intern. Congress Math., Stockholm, 1962, Inst. Mittag-Leffler, 1963, pp. 234-241. MR 31:168

[V1]
V. Voevodsky, Triangulated categories of motives over a field, UIUC K-theory preprint server, no. 74, 1995.

[V2]
-, The Milnor Conjecture, UIUC K-theory preprint server, no. 170, 1996.

[Wg]
J. B. Wagoner, Continuous cohomology and $p$-adic $K$-theory, Lecture Notes in Math., vol. 551, Springer Verlag, 1976, pp. 241-248. MR 58:16609

[Ws]
L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83, Springer Verlag, 1982. MR 85g:11001

[W1]
C. Weibel, Étale Chern classes at the prime $2$, Algebraic $K$-theory and Algebraic Topology (P. Goerss and J. F. Jardine, eds.), NATO ASI Series C, vol. 407, Kluwer, 1993, pp. 249-286. MR 96m:19013

[W2]
-, An introduction to homological algebra, Cambridge Univ. Press, 1994. MR 95f:18001

[W3]
-, The $2$-torsion in the $K$-theory of the integers, C. R. Acad. Sci. Paris 324 (1997), 615-620. MR 98h:19001

[Wi]
A. Wiles, The Iwasawa Conjecture for totally real fields, Annals of Math. 131 (1990), 493-540. MR 91i:11163

References to Appendix A:

[Coa]
J. Coates, $p$-adic $L$-functions and Iwasawa's theory, Algebraic Number Fields (A. Fröhlich, ed.), Academic Press, London, 1977, pp. 269-353. MR 57:276

[DR]
P. Deligne and K. Ribet, Values of Abelian $L$-functions at negative integers over totally real fields, Invent. Math. 59 (1980), 227-286. MR 81m:12019

[Fed]
L. J. Federer, Regulators, Iwasawa modules, and the Main Conjecture for $p=2$, Number Theory Related to Fermat's Last Theorem (N. Koblitz, ed.), Birkhäuser, Basel, 1982, pp. 289-296. MR 84a:10004

[Gr]
R. Greenberg, On $p$-adic $L$-functions and cyclotomic fields, Nagoya Math. J. 56 (1974), 61-77. MR 50:12984

[Gr2]
R. Greenberg, On $p$-adic $L$-functions and cyclotomic fields. II, Nagoya Math. J. 67 (1977), 139-158. MR 56:2964

[Gr3]
R. Greenberg, On $p$-adic Artin $L$-functions, Nagoya Math. J. 89 (1983), 77-87. MR 85b:11104

[Gri]
C. Greither, Class Groups of Abelian Fields, and the Main Conjecture, Ann. Inst. Fourier, Grenoble 42 (1992), 449-499. MR 93j:11071

[Iwa]
K. Iwasawa, Lectures on $p$-adic $L$-functions, Princeton University Press, Princeton, 1972. MR 50:12974

[Iwa2]
K. Iwasawa, On $\mathbb Z_{l}$-extensions of algebraic number fields, Annals of Math. 98 (1973), 246-326. MR 50:2120

[Ko]
M. Kolster, A relation between the $2$-primary parts of the Main Conjecture and the Birch-Tate Conjecture, Can. Bull. Math. 32 (1989), 248-251. MR 90k:11154

[KNF]
M. Kolster, T. Nguyen Quang Do and V. Fleckinger, Twisted $S$-units, $p$-adic class number formulas and the Lichtenbaum Conjectures, Duke Math. J. 84 (1996), 679-717. MR 97g:11136; MR 98k:11172

[Li]
S. Lichtenbaum, On the values of zeta and $L$-functions: I, Annals of Math. 96 (1972), 338-360. MR 50:12975

[MW]
B. Mazur and A. Wiles, Class-fields of Abelian extensions of $\mathbb Q$, Invent. Math. 76 (1984), 179-330. MR 85m:11069

[Ngu]
T. Nguyen Quang Do, Une Étude Cohomologique de la Partie $2$-Primaire de $K_{2}\mathcal{O}$, $K$-Theory 3 (1990), 523-542. MR 91h:11130

[Ws]
L. C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer, 1982. MR 85g:11001

[Wi]
A. Wiles, The Iwasawa conjecture for totally real fields, Annals of Math. 131 (1990), 493-540. MR 91i:11163

[Z]
T. Zink, Étale cohomology and duality in algebraic number fields, Appendix 2, Galois cohomology of algebraic number fields, by K. Haberland, Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 81i:12009

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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
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

DOI: 10.1090/S0894-0347-99-00317-3
PII: S 0894-0347(99)00317-3
Keywords: Two-primary algebraic $K$-theory, number fields, Lichtenbaum--Quillen conjectures, étale cohomology, motivic cohomology, Bloch--Lichtenbaum spectral sequence
Received by editor(s): July 13, 1998
Posted: August 23, 1999
Copyright of article: Copyright 1999, American Mathematical Society

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