Stability for quadratic K-1

Bak A, Petrov V, Tang G. Stability for quadratic K-1. K-Theory. 2003;30(1):1-11.The general quadratic group GQ(2n) and its elementary subgroup EQ(2n) are analogs in the theory of quadratic forms of the general linear group GL(n) and its elementary subgroup E-n. This article proves that the stabilization map GQ(2n)/EQ(2n) --> GQ(2(n+1))/EQ(2(n+1)) is an isomorphism whenever n greater than or equal to LambdaS + 1 and LambdaS denotes the Lambda-stable rank of rings with anti-involution. As a corollary, a result is obtained which has been anticipated since the late 1960s: over rings of finite Bass-Serre dimension d, the stabilization map is an isomorphism whenever n greater than or equal to d + 2