On the Lichtenbaum-Quillen Conjecture

Abstract

The Lichtenbaum-Quillen conjecture, relating the algebraic K-theory of rings of integers in number fields to their etale cohomology, has been one of the main factors of development of algebraic K-theory in the beginning of the 1980s. Soule’ and Dwyer-Friedlander mapped algebraic K-theory of a ring of integers to its l-adic cohomology by means of a ‘Chern character’, that they proved surjective. Here, on the contrary, we map etale 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.