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.
Forster–Swan theorem 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.
Statement Define the local -bound
then the following holds
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