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^[1] and then in 1967 generalized by Richard G. Swan^[2] to its modern form.

Forster–Swan theorem

Let

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

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

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

Statement

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

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

then the following holds^[3]

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

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.