On the number of generators of modules over polynomial affine rings (Q1183064)

The main result of the paper is the following: Let \(A\) be an affine domain and put \(B=A[X_ 1,\ldots,X_ n]\). Let \(M\) be a finitely generated \(B\)-module of \(\hbox{rank} r\). Denote by \(d\) the Krull dimension of \(A\), by \(\mu(M)\) the minimal number of generators of \(M\) and by \(I_ M\) the (radical) ideal which defines the set of primes of \(B\) at which \(M\) is not locally free. Assume that: \(\mu(M/I_ MM)\leq\eta\) and \(\eta\geq\max\{d+r,\dim(B/I_ M)+r+1\}\). Then \(\mu(M)\leq\eta\). As a consequence, we get that: \(\mu(M)\leq\max\{d+r,\mu(M_{\mathfrak p})+\dim B/{\mathfrak p}\}\) where \({\mathfrak p}\) runs over the prime ideals of \(B\) such that \(M_{\mathfrak p}\) is not free. This gives a simultaneous generalization of the Eisenbud-Evans bound, as proved by \textit{A. Sathaye} (to polynomial rings in many variables) and of the Quillen-Suslin theorem (to the non-projective case).