Two-primary algebraic \(K\)-theory of rings of integers in number fields (with an appendix by M. Kolster).

*(English)*Zbl 0934.19001Let \(F\) be a totally real number field with \(r_1\) real embeddings and with ring of integers \(\mathcal O_F\). Write \(R=\mathcal O_F[{1\over 2}]\) for the ring of \(2\)-integers in \(F\). Then for all even \(i>0\) one has
\[
2^{r_1}\cdot{{\# K_{2i-2}(R)\{2\}}\over{\# K_{2i-1}(R)\{2\}}}= {{\# H^2_{\text{ét}}(R;{\mathbb Z}_2(i))}\over{\# H^1_{\text{ét}}(R;{\mathbb Z}_2(i))}},
\]
where, for an abelian group \(A\), \(A\{2\}\) denotes its \(2\)-primary torsion subgroup. This result confirms Lichtenbaum’s conjecture for the \(2\)-primary \(K\)-theory and its relation with étale cohomology of \(F\). This follows from an explicit description of the algebraic \(K\)-groups of \(\mathcal O_F\) (or \(R\)) in terms of étale cohomology, modulo odd finite groups. This is done, using Borel’s famous calculation of \(\dim_{\mathbb Q}(K_n(R)\otimes_{\mathbb Z}{\mathbb Q})\), for all \(n\geq 2\) and \(F\) being a number field with at least one real embedding. Using results of Voevodsky-Suslin and of Bloch-Lichtenbaum, Dwyer-Friedlander theory already gives an explicit expression for \(K_n(R)\) up to an odd finite group for totally imaginary number fields. One also has local analogs, i.e., for a finite extension \(E\) of \({\mathbb Q}_p\) (of characteristic zero), a so-called \(p\)-local field (of characteristic zero), with valuation ring \(\mathcal O_E\) one can describe \(K_n(\mathcal O_E;{\mathbb Z}_2)\simeq K_n(E;{\mathbb Z}_2)\).

Another (partial) result for a long standing conjecture of Lichtenbaum relating the values \(\zeta_F(1-i)\) of the zeta-function of \(F\) to étale cohomology (thus also to \(K\)-theory) can also be stated: Let \(F\) be a totally real abelian number field, then for all even \(i\geq 0\), \[ \zeta_F(1-i)\sim_22^{r_1}\cdot{{\#K_{2i-2}(R)\{2\}}\over{\#K_{2i-1}(R)\{2\}}}, \] where \(\sim_2\) means that both sides are rational numbers having the same \(2\)-adic valuation. This result follows from a theorem of A. Wiles on the Main Conjecture of Iwasawa theory, as explained in an appendix by M. Kolster.

Although the (Quillen-)Lichtenbaum conjectures have undergone much study from their beginnings, the results obtained in the present paper could only be proved using quite recent results of Bloch-Lichtenbaum and of Suslin and Voevodsky. Basic to the setup is the Bloch-Lichtenbaum third quadrant spectral sequence (which holds for any field \(F\)): \[ E_2^{p,q}=CH^{-q}(F,-p-q)\Rightarrow K_{-p-q}(F), \] where \(CH^i(F,n)\) denote Bloch’s higher Chow groups of \(\text{Spec}(F)\). Actually a version with coefficients \({\mathbb Z}/m\) is needed (and proved in an appendix): \[ E_2^{p,q}=CH^{-q}(F,-p-q;{\mathbb Z}/m)\Rightarrow K_{-p-q}(F;{\mathbb Z}/m). \] Then results of Suslin and Voevodsky relate the \(CH^i(F;{\mathbb Z}/m)\) to motivic cohomology \(H_{\mathcal M}\) of \(F\), and then also (in characteristic zero) to étale cohomology \(H_{\text{ét}}\) of \(F\). For the Bloch-Lichtenbaum spectral sequence this leads to: \[ E_2^{p,q}=\begin{cases} H^{p-q}_{\text{ét}}(F;{\mathbb Z}/2^{\nu}(-q)),&q\leq p\leq 0\\ 0,&\text{ otherwise} \end{cases}\Rightarrow K_{-p-q}(F;{\mathbb Z}/2^{\nu}). \] One may pass to the colimit over \(\nu\), then with \(W(i)\) denoting the union of the étale sheaves \({\mathbb Z}/2^{\nu}(i)\), one obtains: \[ E_2^{p,q}=\begin{cases} H^{p-q}_{\text{ét}}(F;W(-q)),&q\leq p\leq 0\\ 0,&\text{ otherwise} \end{cases}\Rightarrow K_{-p-q}(F;{\mathbb Z}/2^{\infty}). \] Several interesting cases follow easily: (i) \(F\) totally imaginary; (ii) \(F=E\) a \(p\)-local field and \(\nu=1\).

The paper consists of: (i) an introduction with some history of Lichtenbaum’s conjecture and a presentation of the results to be proved in subsequent sections; (ii) the Bloch-Lichtenbaum spectral sequence, as described above; (iii) a review of étale cohomology, in particular the description of \(H^0\), \(H^1\) and \(H^2\) of number fields and of \(p\)-local fields; (iv) two-primary algebraic \(K\)-theory of \(2\)-local fields (of characteristic zero); (v) étale cohomology of global fields, in particular various maps between the cohomology of a number field and of its localizations; (vi) the spectral sequence for the real numbers \({\mathbb R}\), in particular \(K_n({\mathbb R};{\mathbb Z}/2)\) and \(K_n({\mathbb R};{\mathbb Z}/2^{\infty})\); (vii) two-primary \(K\)-theory of number fields, describing \(K_n(F;{\mathbb Z}/2^{\infty})\), \(K_n(R;{\mathbb Z}/2^{\infty})\) and \(K_n(R)\{3\}\); (viii) \(\pmod 2\) \(K\)-groups, in particular the \(K_n(R;{\mathbb Z}/2)\). Many interesting side results emerge on the fly. The paper closes with two appendices (and references), the first on the cohomological version of the Lichtenbaum conjecture at the prime \(2\) by M. Kolster, and the second on the Bloch-Lichtenbaum spectral sequence with coefficients. Concluding, one may say that Lichtenbaum’s conjectures are still of great interest and intrigue, and one may hope that new methods will give a further breakthrough.

Another (partial) result for a long standing conjecture of Lichtenbaum relating the values \(\zeta_F(1-i)\) of the zeta-function of \(F\) to étale cohomology (thus also to \(K\)-theory) can also be stated: Let \(F\) be a totally real abelian number field, then for all even \(i\geq 0\), \[ \zeta_F(1-i)\sim_22^{r_1}\cdot{{\#K_{2i-2}(R)\{2\}}\over{\#K_{2i-1}(R)\{2\}}}, \] where \(\sim_2\) means that both sides are rational numbers having the same \(2\)-adic valuation. This result follows from a theorem of A. Wiles on the Main Conjecture of Iwasawa theory, as explained in an appendix by M. Kolster.

Although the (Quillen-)Lichtenbaum conjectures have undergone much study from their beginnings, the results obtained in the present paper could only be proved using quite recent results of Bloch-Lichtenbaum and of Suslin and Voevodsky. Basic to the setup is the Bloch-Lichtenbaum third quadrant spectral sequence (which holds for any field \(F\)): \[ E_2^{p,q}=CH^{-q}(F,-p-q)\Rightarrow K_{-p-q}(F), \] where \(CH^i(F,n)\) denote Bloch’s higher Chow groups of \(\text{Spec}(F)\). Actually a version with coefficients \({\mathbb Z}/m\) is needed (and proved in an appendix): \[ E_2^{p,q}=CH^{-q}(F,-p-q;{\mathbb Z}/m)\Rightarrow K_{-p-q}(F;{\mathbb Z}/m). \] Then results of Suslin and Voevodsky relate the \(CH^i(F;{\mathbb Z}/m)\) to motivic cohomology \(H_{\mathcal M}\) of \(F\), and then also (in characteristic zero) to étale cohomology \(H_{\text{ét}}\) of \(F\). For the Bloch-Lichtenbaum spectral sequence this leads to: \[ E_2^{p,q}=\begin{cases} H^{p-q}_{\text{ét}}(F;{\mathbb Z}/2^{\nu}(-q)),&q\leq p\leq 0\\ 0,&\text{ otherwise} \end{cases}\Rightarrow K_{-p-q}(F;{\mathbb Z}/2^{\nu}). \] One may pass to the colimit over \(\nu\), then with \(W(i)\) denoting the union of the étale sheaves \({\mathbb Z}/2^{\nu}(i)\), one obtains: \[ E_2^{p,q}=\begin{cases} H^{p-q}_{\text{ét}}(F;W(-q)),&q\leq p\leq 0\\ 0,&\text{ otherwise} \end{cases}\Rightarrow K_{-p-q}(F;{\mathbb Z}/2^{\infty}). \] Several interesting cases follow easily: (i) \(F\) totally imaginary; (ii) \(F=E\) a \(p\)-local field and \(\nu=1\).

The paper consists of: (i) an introduction with some history of Lichtenbaum’s conjecture and a presentation of the results to be proved in subsequent sections; (ii) the Bloch-Lichtenbaum spectral sequence, as described above; (iii) a review of étale cohomology, in particular the description of \(H^0\), \(H^1\) and \(H^2\) of number fields and of \(p\)-local fields; (iv) two-primary algebraic \(K\)-theory of \(2\)-local fields (of characteristic zero); (v) étale cohomology of global fields, in particular various maps between the cohomology of a number field and of its localizations; (vi) the spectral sequence for the real numbers \({\mathbb R}\), in particular \(K_n({\mathbb R};{\mathbb Z}/2)\) and \(K_n({\mathbb R};{\mathbb Z}/2^{\infty})\); (vii) two-primary \(K\)-theory of number fields, describing \(K_n(F;{\mathbb Z}/2^{\infty})\), \(K_n(R;{\mathbb Z}/2^{\infty})\) and \(K_n(R)\{3\}\); (viii) \(\pmod 2\) \(K\)-groups, in particular the \(K_n(R;{\mathbb Z}/2)\). Many interesting side results emerge on the fly. The paper closes with two appendices (and references), the first on the cohomological version of the Lichtenbaum conjecture at the prime \(2\) by M. Kolster, and the second on the Bloch-Lichtenbaum spectral sequence with coefficients. Concluding, one may say that Lichtenbaum’s conjectures are still of great interest and intrigue, and one may hope that new methods will give a further breakthrough.

Reviewer: W.W.J.Hulsbergen (Haarlem)

##### MSC:

19D50 | Computations of higher \(K\)-theory of rings |

11R70 | \(K\)-theory of global fields |

11S70 | \(K\)-theory of local fields |

14F20 | Étale and other Grothendieck topologies and (co)homologies |

19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |

14F42 | Motivic cohomology; motivic homotopy theory |

##### Keywords:

two-primary algebraic \(K\)-theory; number fields; Lichtenbaum-Quillen conjectures; étale cohomology; motivic cohomology; Bloch-Lichtenbaum spectral sequence; Lichtenbaum conjecture
PDF
BibTeX
XML
Cite

\textit{J. Rognes} and \textit{C. Weibel}, J. Am. Math. Soc. 13, No. 1, 1--54 (2000; Zbl 0934.19001)

Full Text:
DOI

**OpenURL**

##### References:

[1] | J. F. Adams, On the groups \?(\?). IV, Topology 5 (1966), 21 – 71. · Zbl 0145.19902 |

[2] | Spencer Bloch, Algebraic cycles and higher \?-theory, Adv. in Math. 61 (1986), no. 3, 267 – 304. · Zbl 0608.14004 |

[3] | S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology, Invent. Math. (to appear). · Zbl 0809.14004 |

[4] | Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235 – 272 (1975). · Zbl 0316.57026 |

[5] | Algebraic number theory, Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. |

[6] | William G. Dwyer and Eric M. Friedlander, Algebraic and etale \?-theory, Trans. Amer. Math. Soc. 292 (1985), no. 1, 247 – 280. · Zbl 0581.14012 |

[7] | E. Friedlander and V. Voevodsky, Bivariant cycle cohomology, UIUC K-theory preprint server, no. 75, 1995. · Zbl 1019.14011 |

[8] | Ofer Gabber, \?-theory of Henselian local rings and Henselian pairs, Algebraic \?-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989) Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 59 – 70. · Zbl 0791.19002 |

[9] | 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). · Zbl 0729.11053 |

[10] | B. Harris and G. Segal, \?\? groups of rings of algebraic integers, Ann. of Math. (2) 101 (1975), 20 – 33. · Zbl 0331.18015 |

[11] | Raymond T. Hoobler, When is \?\?(\?)=\?\?\(^{\prime}\)(\?)?, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, Springer, Berlin-New York, 1982, pp. 231 – 244. · Zbl 0491.14013 |

[12] | Uwe Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), no. 2, 207 – 245. · Zbl 0649.14011 |

[13] | Bruno Kahn, Some conjectures on the algebraic \?-theory of fields. I. \?-theory with coefficients and étale \?-theory, Algebraic \?-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. |

[14] | -, The Quillen-Lichtenbaum Conjecture at the prime \(2\), UIUC K-theory preprint server, no. 208, 1997. |

[15] | Stephen Lichtenbaum, On the values of zeta and \?-functions. I, Ann. of Math. (2) 96 (1972), 338 – 360. · Zbl 0251.12002 |

[16] | Stephen Lichtenbaum, Values of zeta-functions, étale cohomology, and algebraic \?-theory, Algebraic \?-theory, II: ”Classical” algebraic \?-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 489 – 501. Lecture Notes in Math., Vol. 342. |

[17] | James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. · Zbl 0433.14012 |

[18] | J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. · Zbl 0613.14019 |

[19] | Yu. P. Nesterenko and A. A. Suslin, Homology of the general linear group over a local ring, and Milnor’s \?-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. · Zbl 0668.18011 |

[20] | Jürgen Neukirch, Class field theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 280, Springer-Verlag, Berlin, 1986. · Zbl 0587.12001 |

[21] | I. A. Panin, The Hurewicz theorem and \?-theory of complete discrete valuation rings, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 763 – 775, 878 (Russian). · Zbl 0614.18009 |

[22] | Daniel Quillen, On the cohomology and \?-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972), 552 – 586. · Zbl 0249.18022 |

[23] | Daniel Quillen, Higher algebraic \?-theory. I, Algebraic \?-theory, I: Higher \?-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 85 – 147. Lecture Notes in Math., Vol. 341. · Zbl 0292.18004 |

[24] | Daniel Quillen, Finite generation of the groups \?\? of rings of algebraic integers, Algebraic \?-theory, I: Higher \?-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 179 – 198. Lecture Notes in Math., Vol. 341. |

[25] | Daniel Quillen, Higher algebraic \?-theory, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 171 – 176. · Zbl 0359.18014 |

[26] | D. Quillen, Letter from Quillen to Milnor on \?\?(\?\?\?\to \?\?^{\?}\to \?\?\?), Algebraic \?-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Springer, Berlin, 1976, pp. 182 – 188. Lecture Notes in Math., Vol. 551. · Zbl 0351.55003 |

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

[28] | J. Rognes and P. A. Østvær, Two-primary algebraic \(K\)-theory of two-regular number fields, Math. Z. (to appear). · Zbl 0943.19002 |

[29] | 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 · Zbl 0919.19003 |

[30] | Peter Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), no. 2, 181 – 205 (German). · Zbl 0421.12024 |

[31] | 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. · Zbl 0423.12016 |

[32] | C. Soulé, \?-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1979), no. 3, 251 – 295 (French). · Zbl 0437.12008 |

[33] | A. Suslin, On the \?-theory of algebraically closed fields, Invent. Math. 73 (1983), no. 2, 241 – 245. , https://doi.org/10.1007/BF01394024 M. Karoubi, Homology of the infinite orthogonal and symplectic groups over algebraically closed fields, Invent. Math. 73 (1983), no. 2, 247 – 250. An appendix to the paper: ”On the \?-theory of algebraically closed fields” by A. Suslin. · Zbl 0514.18009 |

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

[35] | Andrei Suslin, Algebraic \?-theory and motivic cohomology, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 342 – 351. · Zbl 0841.19003 |

[36] | A. A. Suslin and V. Voevodsky, The Bloch-Kato conjecture and motivic cohomology with finite coefficients, UIUC K-theory preprint server, no. 83, 1995. · Zbl 1005.19001 |

[37] | John Tate, Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 288 – 295. |

[38] | V. Voevodsky, Triangulated categories of motives over a field, UIUC K-theory preprint server, no. 74, 1995. · Zbl 1019.14009 |

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

[40] | J. B. Wagoner, Continuous cohomology and \?-adic \?-theory, Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Springer, Berlin, 1976, pp. 241 – 248. Lecture Notes in Math., Vol. 551. · Zbl 0355.18016 |

[41] | Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. · Zbl 0484.12001 |

[42] | Charles Weibel, Étale Chern classes at the prime 2, Algebraic \?-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. · Zbl 0904.19002 |

[43] | Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. · Zbl 0797.18001 |

[44] | Charles Weibel, The 2-torsion in the \?-theory of the integers, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 6, 615 – 620 (English, with English and French summaries). · Zbl 0889.11039 |

[45] | A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493 – 540. · Zbl 0719.11071 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.