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 formula_1 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 formula_2.
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
According to Nakayama's lemma, in order to compute formula_13 one can compute the dimension of formula_14 over the field formula_15, i.e.
formula_16
Statement.
Define the local formula_6-bound
formula_18
then the following holds
formula_19