?WHAT IS... a Syzygy? Roger Wiegand Linear algebra over rings is lots more fun than resolution of the following form (the rather com- over fields. The main reason is that most modules plicated matrices defining the maps between free over a ring do not have bases—that is, their gen- modules need not concern us): erators usually satisfy some nontrivial relations 3 ∂3 6 ∂2 4 ∂1 or “syzygies”. Given a finitely generated R-module 0 → R −→ R −→ R −→ R → M → 0 M R (where is a commutative ring) and a set This time it takes three steps to resolve the mod- z ,...,z of generators, a syzygy of M is an element 1 n ule: The first module of syzygies of M is ker(ε), the (a ,...,a ) ∈ Rn for which a z + ···+ a z =0. 1 n 1 1 n n second (i.e., the module of syzygies of ker(ε)) is The set of all syzygies (relative to the given gen- ker(∂ ), and the third module of syzygies is ker(∂ ). erating set) is a submodule of Rn, called the mod- 1 2 The 0 at the left end of the exact sequence then tells ule of syzygies. Thus the module of syzygies of M 3 us that ∂ maps R isomorphically onto ker(∂ ), is the kernel of the map Rn −→ M that takes the 3 2 that is, the third syzygy of M is free. standard basis elements of Rn to the given set of Hilbert’s famous “Syzygy Theorem”, published generators. in 1890, states that every finitely generated graded Suppose, for example, that R = C[x, y] and module M = ⊕∞ M over the polynomial ring M = R/m, where m is the maximal ideal consist- i=0 i C[x ,...,x ] has a free resolution of length at most ing of polynomials with no constant term. The 1 n n, that is, its nth syzygy is free. (The grading re- module of syzygies is m, which is generated by x spects the action of the variables, in the sense that and y. These elements are not independent, since ⊆ ≤ they satisfy the non-trivial relation (−y)x + xy =0. xj Mi Mi+1 for all i and all j n. The length is one So we look at the module S of syzygies of m, which less than the number of free modules in the reso- is the rank-one free module generated by the ele- lution.) Hilbert used this result to prove that the → ment (−y,x) ∈ R2. Thus after taking syzygies twice function i dimC Mi is, for large i, a polynomial we have “resolved” M and obtained a free module. function of i. The idea here is that many proper- We say that S is the second module of syzygies of ties of modules are easy to compute for free mod- R/m. We can encode this information in the fol- ules and behave well along exact sequences. Thus, lowing exact sequence (where we have written el- if a module has a finite free resolution, one can eas- ements of R2 as columns, so that matrices act on ily read off certain kinds of numerical data. the left): It turns out that every finitely generated mod- −y x [xy] ule M (not necessarily graded) over a polynomial 0 → R −→ R2 −→ R → R/ → 0 m ring R = k[x1,...,xn], where k is a field, has a free For a slightly more complicated example, let resolution of length at most n. By introducing an- R = C[x, y, z] and resolve the module other variable and “homogenizing”, one can obtain M := R/(x3y + z4,xy2,x3 + y3,z5) . Using the soft- a free resolution of length at most n +1. Using ho- ware package MACAULAY 2, one obtains a free mological ideas available in the 1950s, one can then show that the nth syzygy S of M is stably free, p ∼ q Roger Wiegand is professor of mathematics at the that is, S ⊕ R = R for suitable integers p, q. Now University of Nebraska. His email address is one uses “Serre’s Conjecture”, proved indepen- [email protected]. dently by Quillen and Suslin in 1976: projective 456 NOTICES OF THE AMS VOLUME 53, NUMBER 4 modules over polynomial rings (such as the R- ematicians should know about syzygies, cf. David module S above) are free. Cox’s article in the November 2005 Notices. Let’s switch gears, from graded rings to local I will close by mentioning a few open problems. rings. Suppose R is a (commutative, Noetherian) First, as might be suggested by our first example, local ring with maximal ideal m, and let M be a fi- for the simple module R/m over a regular local ring nitely generated R-module. By choosing a minimal R of dimension n, the Betti numbers are binomial n generating set for M, and then a minimal generat- coefficients: bi = i . The “Horrocks conjecture” as- ing set for the first syzygy, and so on, one obtains serts that these are the smallest possible Betti num- a free resolution bers for a module of projective dimension n. A re- lated problem is the “syzygy conjecture”, which b b b ···→R n →···→R 1 → R 0 → M → 0. asserts that if M has projective dimension n then b − b + b −···±b ≥ i for each i<n. The It is not hard to see that the syzygies are uniquely i i+1 i+2 n conjecture is true for rings containing a field [3], determined up to isomorphism (independent of the but it is still open in the general case. Another choice of generators at each stage). We let syzi(M) open question concerns modules of infinite pro- denote the ith syzygy, that is, the image of the map jective dimension: Given a finitely generated mod- Rbi → Rbi−1 . The integers b = b (M) are called the i i ule M over an arbitrary local ring R, are the Betti Betti numbers of M . If, for some h, we have numbers bi(M) eventually nondecreasing? bh(M) =0 but bi(M)=0 for i>h, we say M has projective dimension h. For the ring of formal power Further Reading C series [[x1,...,xn]] , it follows from Hilbert’s [1] L. AVRAMOV, Homological asymptotics of modules over theorem that every finitely generated module M local rings, Commutative Algebra (M. Hochster, has projective dimension at most n. On the other C. Huneke, J. D. Sally, eds.), Mathematical Sciences hand, for the cuspidal ring R := C[[t2,t3]] = Research Institute Publications, vol. 15, 1987. C[[x, y]]/(y2 − x3), the minimal resolution of the [2] D. EISENBUD, The Geometry of Syzygies, Graduate Texts simple module R/(x, y) is easily seen to be infinite, in Mathematics, vol. 229, Springer, 2005. and periodic after one step: [3] E. G. EVANS and P. GRIFFITH, Syzygies, London Mathe- matics Society Lecture Notes Series, vol. 106, 1985. 2 2 y −x yx −xy xy ···→R2 → R2 → R2 2 y −x −xy xy → R2 → R → M → 0 On the other hand, the Betti numbers of the module C[[t3,t4,t5]]/(t3,t4,t5) over the ring C[[t3,t4,t5]] grow exponentially. These examples are symptoms of general be- havior promised by deep results in the represen- tation theory of local rings. First and foremost is the theorem of Auslander, Buchsbaum, and Serre, which says that a local ring R is regular (meaning, in the geometric context, that it corresponds to a smooth point on a variety) if and only if every fi- nitely generated R-module has finite projective di- mension. Next, if R is any ring of the form C[[x1,...,xn]]/(f ), where f is a nonzero element in the square of the maximal ideal, then the minimal resolution of every finitely generated R-module is eventually periodic, of period at most two. To place the last example in context, see Avramov’s survey [1] of the asymptotic behavior of Betti numbers over local rings. The “WHAT IS…?” column carries short (one- or Syzygies and the structure of free resolutions are two-page) nontechnical articles aimed at graduate stu- dents. Each article focuses on a single mathematical currently the subject of intense research, though object rather than a whole theory. The Notices I have mentioned only a few highlights. For an ac- welcomes feedback and suggestions for topics. Mes- count of the use of syzygies in algebraic geometry, sages may be sent to [email protected]. cf. Eisenbud’s book [2]. To see why applied math- APRIL 2006 NOTICES OF THE AMS 457.