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 $R$.References
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
- Jürgen Herzog and Rolf Waldi, A note on the Hilbert function of a one-dimensional Cohen-Macaulay ring, Manuscripta Math. 16 (1975), no. 3, 251–260. MR 384785, DOI 10.1007/BF01164427
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
- 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 265339, DOI 10.2307/2373501
- Judith D. Sally, Numbers of generators of ideals in local rings, Marcel Dekker, Inc., New York-Basel, 1978. MR 0485852 O. Zariski and P. Samuel, Commutative algebra, Lecture Notes in Math., vol. 2, Springer-Verlag, Berlin and New York, 1975.
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