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Two-primary algebraic \(K\)-theory of rings of integers in number fields (with an appendix by M. Kolster). (English) Zbl 0934.19001
Let \(F\) 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.

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
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