On Borel's stable range of the twisted cohomology of $\mathrm{G}\mathrm{L}(n,\mathbb{Z})$

Borel’s stability and vanishing theorem gives the stable cohomology of $\mathrm{G}\mathrm{L}(n,\mathbb{Z})$ with coefficients in algebraic $\mathrm{G}\mathrm{L}(n,\mathbb{Z})$-representations. We compute the improved stable range that Borel remarked about. In order to further improve Borel’s stable range, we adapt the method of Kupers-Miller-Patzt to any algebraic $\mathrm{G}\mathrm{L}(n,\mathbb{Z})$-representations.