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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Presentation of associated graded rings of Cohen-Macaulay local rings
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by Young-Hyun Cho PDF
Proc. Amer. Math. Soc. 89 (1983), 569-573 Request permission

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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Additional Information
  • © Copyright 1983 American Mathematical Society
  • 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