16 - Numbers of Generators of Modules

Summary

We turn in this chapter to a very “classical” problem – estimating the minimum number of generators needed for a finitely generated module A over a noetherian ring R. In case R is commutative, there is a theorem of Forster from 1964 giving an estimate for the number of generators of A in terms of “local data,” namely, the values g(A, P) + K.dim(R/P), where P is any prime ideal of R and g(A, P) is the minimum number of generators of the localized module A_P over the local ring R_P. In the noncommutative case, we shall see that an appropriate analog of g(A, P) is the minimum number of generators needed for the tensor product of A with the Goldie quotient ring of R/P. With this adjustment, we shall derive an analog of Forster's theorem for finitely generated modules over any FBN ring.

In order to handle data from all prime ideals at once, it is most convenient to work topologically. Thus, we first develop an appropriate topology for the prime spectrum of R, and then we develop a continuity theorem for a normalized version of g(A, P) (considered as a function of P). In case R is FBN, this normalized function turns out to be locally constant, and the estimate for the number of generators of A can be obtained without too much further work.