By Winfried Bruns, H. Jürgen Herzog
Some time past 20 years Cohen-Macaulay jewelry and modules were significant issues in commutative algebra. This e-book meets the necessity for an intensive, self-contained advent to the topic. The authors emphasize the research of particular, particular jewelry, making the presentation as concrete as attainable. the overall thought is utilized to a couple of examples and the connections with combinatorics are highlighted. all through each one bankruptcy, the authors have provided many examples and routines.
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Let R be a graded ring and I an ideal generated by elements of positive degree. Let 1 . . n be prime ideals such that I 6 i for i = 1 . . n. Then there exists a homogeneous element x 2 I, x 2= 1 n. L 1 R . Since I is generated by elements of positive Proof. Let S = j =0 j degree, one has I \ S 6 i \ S for i = 1 . . n. Therefore we may assume that R is positively graded. Furthermore it is harmless to replace i by i for all i. 0 . ng and that there is a homogeneous x0 2 nI with x0 2= 1 n;1. If x T2= n, then we are done.
As to the su ciency of (a)(i) and (ii), note that U = 0 for all 2 Ass R by (i), and, by (ii), depth U 1 if depth R 1. It follows that Ass U = , hence U = 0. For the su ciency of (b)(i) and (ii) we may now use that (a) gives us an exact sequence 0 ! M ! M ! C ! 0. If depth R 1, then C = 0 by (i). If depth R 2, then depth M 2 by (ii), and depth M 2 by the inequality above. Therefore depth C 1, and it follows that Ass C = . p p p p p p p p p p p p p p p p Rank. The dimension of a nite dimensional vector space over a eld is given either by the minimal number of generators or by the maximal number of linearly independent elements.
The dual of M is the module HomR (M R ), which we usually denote by M the bidual then is M , and analogous conventions apply to homomorphisms. The bilinear map M M ! R , (x ') 7! '(x), induces a natural homomorphism h : M ! M . We say that M is torsionless if h is injective, and that M is re exive if h is bijective. Some relations between the notions just introduced are given in the exercises. 1. Let R be a Noetherian ring, and M a nite R-module. Then: (a) M is torsionless if and only if (i) M is torsionless for all Ass R, and (ii) depth M 1 for Spec R with depth R 1 (b) M is re exive if and only if (i) M is re exive for all with depth R 1, and (ii) depth M 2 for Spec R with depth R 2.