Partial bases and homological stability of $\operatorname{GL}_{n}(R)$ revisited

TL;DR

This paper revisits the homological stability of general linear groups over certain rings, extending recent results to a broader class of rings with specific algebraic properties.

Contribution

It proves slope-1 homological stability for $ ext{GL}_n(R)$ over rings satisfying invariant basis number and Cohen--Macaulay conditions, generalizing previous work.

Findings

Establishes homological stability with $ ext{Z}[1/2]$-coefficients.

Includes rings of stable rank 1, Euclidean domains, and certain Dedekind domains.

Extends recent stability results to a wider class of rings.

Abstract

Let $R$ be a unital ring satisfying the invariant basis number property, that every stably free $R$-module is free, and that the complex of partial bases of every finite rank free module is Cohen--Macaulay. This class of rings includes every ring of stable rank $1$ (e.g. any local, semi-local or Artinian ring), every Euclidean domain, and every Dedekind domain $\mathcal{O}_S$ of arithmetic type where $|S| > 1$ and $S$ contains at least one non-complex place. Extending recent work of Galatius--Kupers--Randal-Williams and Kupers--Miller--Patzt, we prove that the sequence of general linear groups $\operatorname{GL}_n(R)$ satisfies slope-$1$ homological stability with $\mathbb{Z}[1/2]$-coefficients.