Forster–Swan theorem

The Forster–Swan theorem is a result from commutative algebra that states an upper bound for the minimal number of generators of a finitely generated module over a commutative Noetherian ring. The usefulness of the theorem stems from the fact, that in order to form the bound, one only needs the minimum number of generators of all localizations.
The theorem was proven in a more restrictive form in 1964 by Otto Forster and then in 1967 generalized by Richard G. Swan to its modern form.

Let

be a commutative Noetherian ring with one,
be a finitely generated -module,
a prime ideal of.
and are the minimal number of generators needed to generate the -module and the -module, respectively.

According to Nakayama's lemma, in order to compute one can compute the dimension of over the field, i.e.

Define the local -bound
then the following holds