Unordered resolutions and homological stability for linear groups

TL;DR

This paper introduces a new, more direct proof strategy for homological stability of linear groups, focusing on localized coefficients and revealing connections to longstanding conjectures.

Contribution

It presents a modified proof approach for homological stability of linear groups that simplifies previous methods and explores the role of localization at (n-1)! in relation to conjectures.

Findings

New proof strategy for homological stability

Localization at (n-1)! is crucial in the approach

Connections to Mirzaii's conjectures and Suslin's injectivity conjecture

Abstract

In this paper, we develop a modified proof strategy for homological stability of linear groups, with the general linear groups serving as a primary example. Our arguments are more direct than those in the classical works of Quillen and Suslin--Nesterenko, although they apply only with localized coefficients. The localization at (n-1)! that arises in our approach appears to be closely related to several conjectures of Mirzaii as well as to Suslin's injectivity conjecture.