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 $M_{𝔭}$.

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,
• $𝔭$ a prime ideal of $R$.
• $\mu (M),{\mu }_{𝔭}(M)$ are the minimal die number of generators to generated the $R$-module $M$ respectively the $R_{𝔭}$-module $M_{𝔭}$.

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

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

Statement

Define the local $𝔭$-bound

$b_{𝔭}(M):={\mu }_{𝔭}(M)+\dim (R/𝔭),$

then the following holds

$\mu (M)\le \underset{𝔭}{\sup }\{b_{𝔭}(M)|𝔭\text{is prime},M_{𝔭}\ne 0\}.$

Bibliography

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