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.

💡 Research Summary

The paper presents a streamlined proof of homological stability for the general linear groups GLₙ(F) over an infinite field F, using a novel construction called “unordered resolutions.” Traditional proofs of stability—originating with Quillen’s work on finite fields and later extended by van der Kallen, Nesterenko, and Suslin to infinite fields—rely on intricate spectral sequences derived from group actions on highly structured complexes (often called position complexes). Those approaches yield stability in the range n > i for homology H_i, but the arguments are technically demanding.

The authors introduce a modified chain complex Aₖ generated by all n × k matrices over F, equipped with a left GLₙ(F)‑action and a right Sₖ‑action that incorporates a sign (−1)^{sign(σ)}. By imposing an “i‑type general position condition” on the columns—requiring that any subset of size ≤ i be linearly independent—they obtain a filtration A^{(i)}ₖ ⊂ Aₖ. Quotienting each A^{(i)}ₖ by the Sₖ‑action yields the unordered complexes eA^{(i)}ₖ. The key technical results (Propositions 2.4 and 2.6) show that both A^{(i)}ₖ and eA^{(i)}ₖ are acyclic for every i, with a particularly simple proof for i = n: any cycle can be written as the boundary of a higher‑dimensional element after adding a new vector ã, which exists because the field is infinite.

With these acyclic complexes in hand, the authors apply Quillen’s stability machinery: they consider the spectral sequence I E₁^{p,q}=H_q(GLₙ; eA^{(n)}p) ⇒ H{p+q}(GLₙ; eA^{(n)}_*). Since the complexes are acyclic, the spectral sequence converges to zero. The crucial observation is that after localizing the coefficient ring at (n‑1)! (i.e., working with ℤ