Swan modules over Laurent polynomials

Abstract

Let $\mathrm{\Omega }=P_{n,m}(\mathbb{Z}[C_p])$ and $A=P_{n,m}(\mathbb{Z}),$ where p is a positive integer and $P_{n,m}(R)$ is the R-algebra $P_{n,m}(R)=R[t_1,t_1^{-1},\dots ,t_n,t_n^{-1}]{\otimes }_{R}R[x_1,\dots ,x_m]$. A Swan module is an extension module of the form $0\to I^{(k)}\to X\to A^{(k)}\to 0$, where I is the kernel of the augmentation homomorphism $\mathit{ϵ}:\mathrm{\Omega }\to A$. 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 $\mathbb{Q}({\mathit{\zeta }}_p)$ and ${\mathbb{F}}_p$ is the field with p elements, then the canonical homomorphism ${\mathit{GL}}_k(P_{n,m}(R))\twoheadrightarrow {\mathit{GL}}_k(P_{n,m}({\mathbb{F}}_p))$ is surjective for all $k\ge 1$.