On the Lichtenbaum-Quillen conjecture. (English) Zbl 0885.19004

Kahn, Bruno

Goerss, P. G. (ed.) et al., Algebraic \(K\)-theory and algebraic topology. Proceedings of the NATO Advanced Study Institute, Lake Louise, Alberta, Canada, December 12–16, 1991. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 407, 147-166 (1993).

Summary: The Lichtenbaum-Quillen conjecture, relating the algebraic \(K\)-theory of rings of integers in number fields to their étale cohomology, has been one of the main factors of development of algebraic \(K\)-theory in the beginning of the 1980s. Soulé and Dwyer-Friedlander mapped algebraic \(K\)-theory of a ring of integers to its \(\ell\)-adic cohomology by means of a ‘Chern character’, that they proved surjective. Here, on the contrary, we map étale cohomology to algebraic \(K\)-theory, providing a right inverse to these Chern characters. This gives a different proof of surjectivity, which avoids Dwyer-Friedlander’s use of ‘secondary transfer’. The constructions and results of this paper concern a much wider class of rings than rings of integers in number fields.
For the entire collection see [Zbl 0880.00040].

MSC:

19F27	Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
11R70	\(K\)-theory of global fields
19L10	Riemann-Roch theorems, Chern characters

Keywords:

algebraic \(K\)-theory of rings of integers in number fields; \(\ell\)-adic cohomology; Lichtenbaum-Quillen conjecture; étale cohomology; Chern characters