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{{cite journal |first=Otto |last=Forster |title=Über die Anzahl der Erzeugenden eines Ideals in einem Noetherschen Ring|journal=Mathematische Zeitschrift |volume=84|pages=80–87 |date=1964 |doi=10.1007/BF01112211}} and then in 1967 generalized by Richard G. Swan{{cite journal |first=Richard G. |last=Swan |title=The number of generators of a module |journal=Math. Mathematische Zeitschrift |volume=102 |date=1967 |issue=4 |pages=318–322 |doi=10.1007/BF01110912 |url=https://eudml.org/doc/170878}} 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{{citation|author=R. A. Rao und F. Ischebeck |date=2005 |editor=Physica-Verlag |location=Deutschland |pages=221 |title=Ideals and Reality: Projective Modules and Number of Generators of Ideals}}

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

Bibliography

{{cite book |first1=R.A. |last1=Rao |first2=F. |last2=Ischebeck |date=2005 |title=Ideals and Reality: Projective Modules and Number of Generators of Ideals |place=Deutschland |publisher=Physica-Verlag}}
{{cite journal |first=Richard G. |last=Swan |title=The number of generators of a module |journal=Math. Mathematische Zeitschrift |volume=102 |date=1967 |issue=4 |pages=318–322 |doi=10.1007/BF01110912 |url=https://eudml.org/doc/170878}}

References

Category: Commutative algebra

Category:Theorems in algebra