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 ${\displaystyle M}${\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 need the minimum number of generators of all localizations ${\displaystyle M_{\mathfrak {p}}}${\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.

Let

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

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

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

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

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

then the following holds^[3]

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

• 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.

• Commutative algebra

• CS1 maint: location missing publisher