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 need 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.

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

Statement

Define the local ${\mathfrak {p}}$-bound

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

then the following holds

$\mu (M)\leq \sup _{\mathfrak {p}}\;\{b_{\mathfrak {p}}(M)\;|\;{\mathfrak {p}}\;{\text{is prime}},\;M_{\mathfrak {p}}\neq 0\}.$

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.