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 $M$ 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 $M_{\mathfrak{𝔭}}$.

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.

Forster–Swan theorem

Let

• $R$ be a commutative Noetherian ring with one,
• $M$ be a finitely generated $R$-module,
• $\mathfrak{𝔭}$ a prime ideal of $R$.
• $\mu (M),{\mu }_{\mathfrak{𝔭}}(M)$ are the minimal die number of generators to generated the $R$-module $M$ respectively the $R_{\mathfrak{𝔭}}$-module $M_{\mathfrak{𝔭}}$.

According to Nakayama's lemma, in order to compute ${\mu }_{\mathfrak{𝔭}}(M)$ one can compute the dimension of $M_{\mathfrak{𝔭}}/\mathfrak{𝔭}M$ over the field $k(\mathfrak{𝔭})=R_{\mathfrak{𝔭}}/\mathfrak{𝔭}{R}_{\mathfrak{𝔭}}$, i.e.

${\mu }_{\mathfrak{𝔭}}(M)={\dim }_{k(\mathfrak{𝔭})}(M_{\mathfrak{𝔭}}/\mathfrak{𝔭}M).$

Statement

Define the local $\mathfrak{𝔭}$-bound

$b_{\mathfrak{𝔭}}(M):={\mu }_{\mathfrak{𝔭}}(M)+\dim (R/\mathfrak{𝔭}),$

then the following holds

$\mu (M)\le \underset{\mathfrak{𝔭}}{\sup }\{b_{\mathfrak{𝔭}}(M)|\mathfrak{𝔭}\text{is prime},M_{\mathfrak{𝔭}}\ne 0\}.$

Bibliography

• Rao, R.A.; Ischebeck, F. (2005). Ideals and Reality: Projective Modules and Number of Generators of Ideals. Deutschland: Physica-Verlag. 
• Swan, Richard G. (1967). "The number of generators of a module". Math. Mathematische Zeitschrift 102 (4): 318–322. doi:10.1007/BF01110912. https://eudml.org/doc/170878.