Unstable analogues of the Lichtenbaum-Quillen conjecture

Abstract

This survey is mostly concerned with unstable analogues of the Lichtenbaum-Quillen Conjecture. The Lichtenbaum-Quillen Conjecture (now implied by the Voevodsky-Rost Theorem) attempts to describe the algebraic K-theory of rings of integers in number fields in terms of much more accessible "etale models". Suitable versions of the conjecture predict the cohomology of infinite general linear groups of rings of S-integers over suitable number fields; our survey focuses on an unstable version of this form of the conjecture.