Forster–Swan theorem

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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 {\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 needs the minimum number of generators of all localizations M p {\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.

Forster–Swan theorem

Let

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

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

μ p ( M ) = dim k ( p ) ⁡ ( M p / p M ) . {\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).}

Statement

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

b p ( M ) := μ p ( M ) + dim ⁡ ( R / p ) , {\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]

μ ( M ) ≤ sup p { b p ( M ) | p is prime , M p ≠ 0 } . {\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\}.}

Bibliography

References

  1. Forster, Otto (1964). "Über die Anzahl der Erzeugenden eines Ideals in einem Noetherschen Ring". Mathematische Zeitschrift. 84: 80–87. doi:10.1007/BF01112211.
  2. Swan, Richard G. (1967). "The number of generators of a module". Math. Mathematische Zeitschrift. 102 (4): 318–322. doi:10.1007/BF01110912.
  3. R. A. Rao und F. Ischebeck (2005), Physica-Verlag (ed.), Ideals and Reality: Projective Modules and Number of Generators of Ideals, Deutschland, p. 221{{citation}}: CS1 maint: location missing publisher (link)