Maximal Cohen–Macaulay Modules over Gorenstein Rings and Bourbaki-Sequences
暂无分享,去创建一个
Motivated by recent work of Knorrer [17], Buchweitz, Greuel and Schreyer [9] who proved that a hypersurface singularity R over C is of finite Cohen-Macaulay representation type if and only if Risa simple hypersurface singularity, we were led to study, quite in general, maximal CohenMacaulay modules (MCM-modules) over Gorenstein rings. Of course, there is no deeper reason why one should restrict one's attention to Gorenstein rings. It seems, however, that representation theory over nonGorenstein rings is fundamentally more complicated. For instance, the question of the finite Cohen-Macaulay representation type is not completely settled, though some beautiful techniques have been developed just for that purpose by M. Auslander and I. Reiten [20]. They also give two examples of non-Gorenstein rings of dim > 3, which are of finite CohenMacaulay representation type. No other such examples are known. In [14] the first named author of the paper showed that C[ x, y] 0 is of finite Cohen-Macaulay representation type, where G<:;;;_G/(2; C) is finite. That these are the only 2-dimensional rings with this property was shown by Artin-Verdier [2], Auslander [3] and Esnault [13]. The great technical advantage of Gorenstein rings is that MCMmodules over Gorenstein rings are reflexive, and that the R-dual of an MCM-module is again an MCM-module. For the rest of the paper let us always assume that (R, m) is a local Gorenstein ring. The most general question one may raise in this connection is to determine all isomorphism classes of indecomposable MCMmodules over R. Of course, this problem is posed far too generally, and should be considered only as a "Leitmotiv". The following problem seems to be more accessible: Determine all pairs of numbers (m, n) for which there exists an MCM-module M which has rank m and is minimally generated by n elements. We call (m, n) the data of M. In [10] D. Eisenbud gives a very explicit description of the MCM-
[1] F. Schreyer,et al. Cohen-Macaulay modules on hypersurface singularities II , 1987 .