Unstable analogues of the Lichtenbaum-Quillen conjecture

TL;DR

This survey explores unstable analogues of the Lichtenbaum-Quillen Conjecture, focusing on their implications for understanding the cohomology of infinite general linear groups over rings of S-integers.

Contribution

It provides a comprehensive overview of unstable versions of the conjecture and their potential to relate algebraic K-theory to more accessible cohomological models.

Findings

Unstable analogues extend the classical conjecture to broader contexts.

Connections between unstable conjectures and cohomology of linear groups are examined.

The survey highlights open problems and future research directions.

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.