Stably free modules of `big rank' over polynomial rings are free (Q914758)

Let A be a commutative ring with \(\dim (A[x_ 1,...,x_ n])=r\), P be a module over \(A[x_ 1,...,x_ n]\) with rank\((P)=\rho\). The authors prove that \[ (\exists d)(\rho >d\text{ and P is stably free }\Rightarrow P\text{ is free}) \] with \(d=r-n\). If A is Noetherian, this result is well-known with \(d=\dim (A)\) [\textit{T. Y. Lam}, ``Serre's Conjecture'', Lect. Notes Math. 635 (1978; Zbl 0373.13004)].