Swan modules over Laurent polynomials

Let Ω=Pn,m(Z[Cp]) and A=Pn,m(Z), where p is a positive integer and Pn,m(R) is the R-algebra Pn,m(R)=R[t1,t1−1,…,tn,tn−1]⊗RR[x1,…,xm]. A Swan module is an extension module of the form 0→I(k)→X→A(k)→0, where I is the kernel of the augmentation homomorphism ϵ:Ω→A. We show that, when p is prime, every such projective Swan module is free; this is false if p is not prime and n+m>0. The proof relies on the fact that when R is the ring of algebraic integers in Q(ζp) and Fp is the field with p elements, then the canonical homomorphism GLk(Pn,m(R))↠GLk(Pn,m(Fp)) is surjective for all k≥1.