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Presentation of associated graded rings of Cohen-Macaulay local rings


Author: Young-Hyun Cho
Journal: Proc. Amer. Math. Soc. 89 (1983), 569-573
MSC: Primary 13H10
DOI: https://doi.org/10.1090/S0002-9939-1983-0718974-8
MathSciNet review: 718974
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Abstract: Let $ (R,\mathfrak{m})$ be a local ring and $ I$ be an $ \mathfrak{m}$-primary ideal such that $ {\dim _k}(I/I\mathfrak{m}) = l$, where $ k = R/\mathfrak{m}$. Denote the associated graded ring with respect to $ I, \oplus _{n = 0}^\infty {I^n}/{I^{n + 1}}$, by $ {G_I}(R)$. Then $ {G_I}(R) \simeq R/I[{X_1}, \ldots {X_l}]/\mathcal{L}$, for some homogeneous ideal $ \mathcal{L}$. Set $ M = \max {\deg _{1 \leqslant i \leqslant t}}{f_i}$, where $ \{ {f_1}, \ldots ,{f_t}\} $ is a set of homogeneous elements which form a minimal basis of $ \mathcal{L}$. The main result in this note is that if $ R$ is a Cohen-Macaulay local ring of dimension 1 and if $ {G_I}(R)$ is free over $ R/I$, then $ M \leqslant r(I) + 1$, where $ r(I)$ is the reduction number of $ I$. It follows that $ M \leqslant e(R)$ where $ e(R)$ is the multiplicity of $ R$.


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  • [1] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. II, Ann. of Math. (2) 79 (1964), 205-326. MR 0199184 (33:7333)
  • [2] J. Herzog and R. Waldi, A note on the Hilbert function of a $ 1$-dimensional Cohen-Macaulay ring, Manuscripta Math. 16 (1975), 251-260. MR 0384785 (52:5658)
  • [3] H. Matsumura, Commutative algebra, 2nd ed., Benjamin Cummings, New York, 1980. MR 575344 (82i:13003)
  • [4] L. J. Ratliff, Jr., On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals. II, Amer. J. Math. 92 (1970), 99-144. MR 0265339 (42:249)
  • [5] J. D. Sally, Numbers of generators of ideals in local rings, Lecture Notes in Pure and Appl. Math., Dekker, New York and Basel, 1978. MR 0485852 (58:5654)
  • [6] O. Zariski and P. Samuel, Commutative algebra, Lecture Notes in Math., vol. 2, Springer-Verlag, Berlin and New York, 1975.

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DOI: https://doi.org/10.1090/S0002-9939-1983-0718974-8
Article copyright: © Copyright 1983 American Mathematical Society

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