The Steinberg group of a polynomial ring

The main result of this paper is the following
Theorem. {\it If $A$ is a Noetherian ring$,$ then the canonical homomorphism $K_{2, r}(A[x_1,\dots,x_n])\to K_2(A[x_1,\dots,x_n])$ is surjective when $r\geqslant\max(4,\dim A+2)$ and injective when $r\geqslant\max(5,\dim A+3)$.}
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