A note on residual variables of an affine fibration

In a recent paper [7], M.E. Kahoui has shown that if R is a polynomial ring over C, A an A3-fibration over R, and W a residual variable of A then A is stably polynomial over R[W]. In this article we show that the above result holds over any Noetherian domain R provided the module of differentials ΩR(A) of the affine fibration A (which is necessarily a projective A-module by a theorem of Asanuma) is a stably free A-module.