Syzygies of Small Rank by E. Graham Evans and Phillip Griffith

For M an R-module and m an element of M we define the order ideal, 0M(m), to be the ideal which consists of the images of m under homomorphisms of M to R. The proof proceeds by considering the height of particular order ideals, 0M(m), in relation to the projective dimension of M. Specifically, if M is a kth syzygy of the 7?-module TV and if xx, . . . ,xk_l is an R sequence, then Torf (TV, R/(xi, . . . , xk__l)) is zero. However, if 0M(m) is contained in the ideal (xx, . . . ,xk_1)9 then considering the map from M to the free module F which occurs in the resolution of TV, the image of m is contained in (xt,. . . , xk_X)F. Thus Torf (TV, R/(x1, . . . , xk_l)) is nonzero. However, if 0M(m) is contained in the ideal (xt, . . . , xk_l), then considering the map from M to the free module F which occurs in the resolution of TV, the image of m is contained in (xx, . . . ,xk_1)F. Thus Tor̂ (TV, R/(xl,... ,xk_x)) is nonzero. This idea is the key step in Gróbner's proof of Hubert's syzygy theorem [4]. It shows that the ideal 0M(m) tends to have height at least k if M is a kth syzygy. On the other hand the height of 0M(m) tends to be bounded by the rank of M. While modules of small rank can have elements with the height of 0M(m) large, we show that such phenomena cannot occur in the minimal counterexample to the syzygy problem. The hypothesis that R contains a field is needed to insure the existence of maximal Cohen-Macaulay modules [6] which replace the module R/(x1,. . . ,xk_l) in our modification of Gróbner's proof [4]. The hypothesis that R is CohenMacaulay is needed to apply the Auslander-Bridger criterion [1] for a module to