By Winfried Bruns, H. Jürgen Herzog
Long ago 20 years Cohen-Macaulay earrings and modules were principal themes in commutative algebra. This ebook 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 final concept is utilized to a few examples and the connections with combinatorics are highlighted. all through every one bankruptcy, the authors have provided many examples and workouts.
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Extra info for Cohen-Macaulay Rings
R by f (x1 x2) = f1 (x1) + f2(x2 ). 6.
7. Let R be a Noetherian ring, and I 6= 0 an ideal with a Proof. nite free resolution. Then I contains an R-regular element. 5. Since I is torsion-free and non-zero, the only possibility is rank I = 1, whence rank R=I = 0. Thus R=I is annihilated by an R -regular element. Proof. Ideals of minors and Fitting invariants. Let U be an m n matrix over R where m n 0. For t = 1 . . min(m n) we then denote by It(U ) the ideal generated by the t-minors of U (the determinants of t t submatrices). For systematic reasons one sets It(U ) = R for t 0 and It(U ) = 0 for t > min(m n).
If one has M = N + 0 M for a graded submodule N of M , then M = N . 25. Let R be a Noetherian positively graded ring, and M a nite graded R -module. Prove dim M = supfdim M : 2 Supp M gradedg. 7. 26. Let R be a graded ring, x a non-nilpotent element of degree 1, A = R=(x ; 1), and : R ! A the natural homomorphism. 18 we set S = Rx , and identify A and S0 . (a) One has (I ) = IS \ A for every homogeneous ideal I R and J = (JS \ R ) for every ideal J of A. ) (b) The homomorphism induces a bijective correspondence between the set of homogeneous ideals of R modulo which x is regular and the set of all ideals of A.