ON THE NUMBER OF GENERATORS OF MODULES OVER POLYNOMIAL-RINGS

In this paper we prove the following Theorem. Let B = A[X1 , ., X(n)], where A is a universally catenary equidimensional ring. Let M be a finitely generated B-module of 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(M)M) less-than-or-equal-to eta and eta greater-than-or-equal-to max{d + r, dim B/I(M) + r + 1}, where eta is a positive integer. Then mu(M) less-than-or-equal-to eta. This improves a result of R. G. Lopez, On the number of generators of modules over polynomial affine rings, Math. Z. 208 (1991), 11-21.