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

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