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Parametric decomposition of powers of ideals versus regularity of sequences


Authors: Shiro Goto and Yasuhiro Shimoda
Journal: Proc. Amer. Math. Soc. 132 (2004), 929-933
MSC (2000): Primary 13H99
DOI: https://doi.org/10.1090/S0002-9939-03-07160-0
Published electronically: October 29, 2003
MathSciNet review: 2045406
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $Q = (a_{1}, a_{2}, \cdots , a_{s}) (\subsetneq A)$ be an ideal in a Noetherian local ring $A$. Then the sequence $a_{1}, a_{2}, \cdots , a_{s}$ is $A$-regular if every $a_{i}$ is a non-zerodivisor in $A$ and if $Q^{n} = \bigcap _{\alpha } (a_{1}^{\alpha _{1}}, a_{2}^{\alpha _{2}}, \cdots , a_{s}^{\alpha _{s}})$ for all integers $n \geq 1$, where $\alpha = (\alpha _{1}, \alpha _{2}, \cdots , \alpha _{s})$ runs over the elements of the set $\Lambda _{s,n} = \{(\alpha _{1}, \alpha _{2}, \cdots , \alpha _{s}) \in {\mathb... ...{for all} 1 \leq i \leq s \ \text{and} \sum _{i=1}^{s}\alpha _{i} = s + n - 1\}$.


References [Enhancements On Off] (What's this?)

  • [GS] S. Goto and Y. Shimoda, On the parametric decomposition of powers of parameter ideals in a Noetherian local ring, Tokyo J. Math., to appear.
  • [HRS] W. Heinzer, L. J. Ratliff, and K. Shah, Parametric decomposition of monomial ideals $(I)$, Houston J. Math. 21 (1995), 29-52. MR 96c:13002
  • [VV] P. Valabrega and G. Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93-101. MR 80d:14010

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Additional Information

Shiro Goto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 214-8571 Japan
Email: goto@math.meiji.ac.jp

Yasuhiro Shimoda
Affiliation: Department of Mathematics, Faculty of General Education, Kitasato University, 228-8555 Japan
Email: shimoda@clas.kitasato-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-03-07160-0
Keywords: Parametric decomposition, regular sequence, grade, Cohen-Macaulay local ring
Received by editor(s): May 28, 2002
Published electronically: October 29, 2003
Additional Notes: The first author is supported by the Grant-in-Aid for Scientific Research in Japan (C(2), No. 13640044)
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2003 American Mathematical Society

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