The Forster–Swan theorem is a result from

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

that states an upper bound for the minimal number of generators of a

finitely generated module In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts i ...

M

over a commutative

Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...

. 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_

. 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, *

\mathfrak

prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...

R

. *

\mu(M),\mu_(M)

are the minimal die number of generators to generated the

R

-module

M

respectively the

R_

-module

M_

. According to

Nakayama's lemma In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and ...

, in order to compute

\mu_(M)

one can compute the dimension of

M_/\mathfrakM

over the field

k(\mathfrak)=R_/\mathfrakR_

, i.e. :

\mu_(M)=\operatorname_(M_/\mathfrakM).

Statement

Define the local

\mathfrak

-bound :

b_(M):=\mu_(M)+\operatorname(R/\mathfrak),

then the following holds :

\mu(M)\leq \sup_\;\.

Bibliography

* *{{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, url-access=subscription

References

Commutative algebra Theorems in algebra