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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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