×

The Lichtenbaum-Quillen conjecture for \(K/\ell_*[\beta^{-1}]\). (English) Zbl 0563.14009

Current trends in algebraic topology, Semin. London/Ont. 1981, CMS Conf. Proc. 2, No. 1, 117-139 (1982).
[For the entire collection see Zbl 0538.00016.]
The main result is the following theorem. Let \(\ell^{\nu}\) be a prime power and X be a regular Noetherian separated scheme of finite Krull dimension. Suppose \(\ell\) is invertible in \({\mathcal O}_ X\) and that all residue fields of X have bounded étale cohomological dimension for \(\ell\)-torsion sheaves. If \(\ell =2\) or 3, assume that X contains enough roots of unity. There is a strongly converging spectral sequence \[ E_ 2^{p,q}=H^ p_{et}(X,{\mathbb{Z}}/\ell^{\nu}(i))\quad for\quad q=2i,\quad E_ 2^{p,q}=0\quad for\quad other\quad q \] converging to \((K/\ell^{\nu})_{q-p}(X)[\beta^{-1}]\) with differentials of bidegree (r,r-1). Here \([\beta^{-1}]\) is a localization by inversion of the Bott element \(\beta\). This has a lot of consequences. In particularly for affine curves X over a finite field we have \[ | \# K_{2i- 2}(X)[\beta^{-1}]^ 1_{\ell}/\# K_{2i-1}(X)[\beta^{-1}]^ 1_{\ell}|_{\ell}=| \zeta (X,1-i)|_{\ell},\quad i\geq 2, \] for \(\ell\)-adique valuations \(| \quad |_{\ell}.\) Here \(K_*(X)[\beta^{-1}]^ 1_{\ell}\) is the homotopy inverse limit of the system \((K/\ell^{\nu})_*(X)[\beta^{-1}]\). There is also the corresponding result for the spectrum of integers of a number field and new Gysin sequences of \({\mathbb{Q}}^ 1_{\ell}\)-cohomologies.
Reviewer: A.N.Parshin

MSC:

14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
14F20 Étale and other Grothendieck topologies and (co)homologies

Citations:

Zbl 0538.00016