On the General Linear Group over Polynomial Rings

Within algebraic K-theory, when A↠B is a ring homomorphism, it is frequently important to determine the image of the induced homomorphism GLk(A)→GLk(B). We show that, if A=A[t,t−1,x] is a ring of mixed polynomials and Laurent polynomials over a commutative ring A of Krull dimension 1 and B=F[t,t−1,x] is the corresponding ring over a finite quotient ring F of A, then SLk(A)→SLk(B) is surjective for all k≥2, provided every stably free A-module is free.