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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
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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Article copyright: © Copyright 1983 American Mathematical Society

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