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We prove that if k is a finite field with $p^d$ elements, then the natural map $H_i(GLₙ\left(k\right),\mathbb{Z})\to H_i(GLₙ\left(k\left[t\right]\right),\mathbb{Z})$ is an isomorphism for 0 ≤ i < d(p-1) and for all n.