View source for Forster-Swan theorem

''Swan–Forster theorem''

An example of a local-global principle in commutative algebra (cf. also [[Local-global principles for large rings of algebraic integers|Local-global principles for large rings of algebraic integers]]; [[Local-global principles for the ring of algebraic integers|Local-global principles for the ring of algebraic integers]]). That is, it provides a method by which local information can be lifted to the whole ring or module. In the case of the Forster–Swan theorem the information consists of the minimal number of elements required to generate a given finitely-generated module. The theorem itself, and also the methods developed in proving it, have found applications in commutative algebra, algebraic geometry and algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f1301601.png" />-theory; see [[#References|[a5]]] and [[#References|[a6]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f1301602.png" /> be a [[Commutative ring|commutative ring]], and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f1301603.png" /> be a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f1301604.png" />-module (cf. also [[Module|Module]]). Suppose that one wants to compute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f1301605.png" />, the minimal number of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f1301606.png" />. This can be a difficult task in general, but one can easily compute the number of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f1301607.png" /> locally. Thus, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f1301608.png" /> be a [[Prime ideal|prime ideal]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f1301609.png" />, and denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016010.png" /> the quotient ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016011.png" />. By the Nakayama lemma (cf. also [[Jacobson radical|Jacobson radical]]),

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

This number will be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016013.png" />, and it will be called the local number of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016014.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016015.png" />. Note that the right-hand side of (a1) is equal to the [[Dimension|dimension]] of a [[Vector space|vector space]], and as such is easily computed. Hence, an upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016016.png" /> in terms of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016017.png" />, for the various prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016019.png" />, would be a very desirable result.

In order to have a means of guessing what upper bound one might expect, one turns to the equivalence between projective modules and fibre bundles. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016020.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016021.png" />-dimensional [[CW-complex|CW-complex]] and denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016022.png" /> its ring of continuous functions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016023.png" /> is a real vector bundle of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016024.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016025.png" />, then there exists a [[Vector bundle|vector bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016027.png" /> is a trivial bundle. Moreover, one can assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016028.png" /> has dimension smaller than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016029.png" />; see [[#References|[a4]]], Chapt. 8, Thms. 1.2, 1.5. Thus, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016030.png" />-module of sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016031.png" /> is isomorphic to a free module of rank at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016032.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016033.png" /> is a homomorphic image of this free module, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016034.png" /> elements must be sufficient to generate it.

Using the above as a guide, one may now return to the general algebraic setting. First, the rank of a projective module at a prime is equal to its minimal number of generators. Secondly, the role of the dimension of the topological space will be played, in the algebraic setting, by the Krull dimension of the base ring (cf. also [[Dimension|Dimension]]). Finally, taking into account the fact that the local number of generators of a module need not be the same at every prime, the topological analogy suggests that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016035.png" /></td> </tr></table>

Assuming that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016036.png" /> is a commutative [[Noetherian ring|Noetherian ring]], this is just what R.O. Forster proved in [[#References|[a3]]]. Following a suggestion of J.-P. Serre, Forster's result was later generalized by R.G. Swan to rings whose maximal spectrum is Noetherian, [[#References|[a9]]] (cf. also [[Spectrum of a ring|Spectrum of a ring]]). Denoting by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016038.png" /> the set of prime ideals that are intersections of maximal ideals, the Forster–Swan theorem can be stated as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016039.png" /> be a commutative ring and assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016040.png" /> is a Noetherian space. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016041.png" /> is a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016042.png" />-module, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016043.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016044.png" /></td> </tr></table>

A number of further improvements are possible. For instance, one can discard all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016046.png" />-primes outside the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016047.png" />, and the Krull dimension can be replaced by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016049.png" />-dimension. For details on these and other improvements, see [[#References|[a2]]] or [[#References|[a5]]], where a proof of the theorem is given based on the notion of a  "basic element" . These methods also produce a similar bound for the stable rank of a module; see [[#References|[a2]]], Corollary 6.

Given the importance of the Forster–Swan theorem, it was natural to ask whether it could be generalized to non-commutative rings. The obstacle was the usual problem of localizing at a prime in a non-commutative algebra. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016050.png" /> is a prime ideal of a Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016052.png" /> is a prime Noetherian ring. Hence it must have a quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016053.png" /> by Goldie's theorem. Therefore, the right-hand side of (a1) makes sense, and there is still a good definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016054.png" /> in this case. This led J.T. Stafford, building on earlier work of R.B. Warfield, to a proof of the Forster–Swan theorem for right and left Noetherian rings [[#References|[a7]]]. A simpler proof, that also works for right Noetherian rings, can be found in [[#References|[a1]]]. For applications of the Forster–Swan theorem to non-commutative algebra, see [[#References|[a8]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.C. Coutinho,   "Generating modules efficiently over noncommutative noetherian rings"  ''Trans. Amer. Math. Soc.'' , '''323'''  (1991)  pp. 843–856</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Eisenbud,   E.G. Evans, Jr.,   "Generating modules efficiently: Theorems from algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016055.png" />-theory"  ''J. Algebra'' , '''27'''  (1973)  pp. 278–305</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Forster,   "Über die Anzahl der Erzeugenden eines Ideals in einem Noetherschen Ring"  ''Math. Z.'' , '''84'''  (1964)  pp. 80–87</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Husemoller,   "Fibre bundles" , Springer  (1966)  (Edition: Second)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Kunz,   "Introduction to commutative algebra and algebraic geometry" , Birkhäuser  (1985)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.C. McConnell,   J.C. Robson,   "Noncommutative noetherian rings" , ''Ser. in Pure and Applied Math.'' , Wiley  (1987)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.T. Stafford,   "Generating modules efficiently: algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130160/f13016056.png" />-theory for noncommutative noetherian rings"  ''J. Algebra'' , '''69'''  (1981)  pp. 312–346</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.T. Stafford,   "The Goldie rank of a module" , ''Noetherian rings and their applications (Oberwolfach, 1983)'' , ''Math. Surveys and Monographs'' , '''24''' , Amer. Math. Soc.  (1987)  pp. 1–20</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R.G. Swan,   "The number of generators of a module"  ''Math. Z.'' , '''102'''  (1967)  pp. 318–322</TD></TR></table>