A polynomial bound on the number of comaximal localizations needed in order to make free a projective module

Let A be a commutative ring and M be a projective module of rank k with n generators. Let h=n-k. Standard computations show that M becomes free after localizations in comaximal elements (see Theorem 5). When the base ring A contains a field with at least hk+1 non-zero distinct elements we construct a comaximal family G with at most (hk+1)(nk+1) elements such that for each g∈G, the module Mg is free over A[1/g].