Minimal number of generators of modules (Q1195945)

This paper deals with conditions on the pair of rings \(R_ 1\subset R_ 2\) under which the minimal number of generators of an \(R_ 1\)-module remains the same when the base is extended to \(R_ 2\). -- In theorem 1.3 it is shown if \(R_ 2\) is faithfully flat over \(R\) and is a ``unity dense'' extension then there is stability. Here \(R_ 1\subset R_ 2\) is said to be ``unity dense'' if for each linear polynomial \(1+\alpha T\in R_ 2[T]\), to each \(r_ 2\in R_ 2\) there is an \(r_ 1\in R_ 1\) such that \(1+\alpha(r_ 1-r_ 2)\) is a unit in \(R_ 2\). -- Theorem 2.2 gives stability under localization. Explicitly, it states that if \(K\) is a nonarchimedean discrete valued field, with a finite residue class field, \(S\) is a fixed subset of a quasiprojective variety \(X\) in \(K^ n\) and \(R=\{f/g\) where \(f,g\) are polynomials from \(K[X_ 1,\ldots,X_ n]\) restricted to \(X\) and \(g(x)\neq 0\) for all \(x\in S\}\) then (a) when \(S\) is a compact subset, then each finitely generated \(R\)-module has the property that the minimal number of generators for \(M\) is equal to the maximum of the minimal number of generators \(M_ m\) -- considered as an \(R_ m\) module as \(m\) varies over the maximal spectrum of \(R\). (b) when \(\dim X=1\) and \(K\) is complete with respect to the valuation then the above assertion holds for \(S=X\) as well. In theorem 2.3, an analogous situation is studied with \(K\) replaced by the algebraic closure of the field of rational numbers \(\mathbb{Q}\) using theorem 1.3. As an application of the above theorem (1.3) the author considers a noetherian ring \(R\) and its \(J\)-adic completion \(\hat R\) and shows the minimal number of generators of \(J\) as an ideal of \(R\) is at most one more than the minimal number of generators of \(J\hat R\) and they are equal if the minimal number exceeds the Krull dimension of \(R\). In case \(J\) is contained in the Jacobson radical of \(R\) then for any \(R\)-module \(M\), stability holds.