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 ${\displaystyle M}$ 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 ${\displaystyle M_{\mathfrak {p}}}$.

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.