\(n\)-stability of a support on a ring and application to projective modules (Q6601235)

The notion of \(n\)-stable support on a ring was coined by Thierry Coquand in order to obtain a constructive proof of Bass-Simis Vasconcelors theorem asserting that for a valuation ring \(R\), every finitely-generated projective \(R[X]\)-module is free. Mainly with the aid of this notion, the paper shows that under some conditions, for any finitely-generated projective \(R[X_1,\dots, X_n]\)-module \(M\) of rank \(r\geq n+1\), \(M\) is isomorphic to a direct sum of a free module of rank \(r-n\) with a module \(N\), where \(N\) is isomorphic to the image of a matrix of rank \(n\).