Unordered resolutions and homological stability for linear groups

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.