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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: 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 $\Lambda _{s,n} = \{(\alpha _{1}, \alpha _{2}, \cdots , \alpha _{s}) \in {\mathbb {Z}}^{s} \mid \alpha _{i} \geq 1 \ \text {for all} \ 1 \leq i \leq s \ \text {and} \ \sum _{i=1}^{s}\alpha _{i} = s + n - 1\}$.


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Additional Information

Shiro Goto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 214-8571 Japan
MR Author ID: 192104
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

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