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

- S. Goto and Y. Shimoda,
*On the parametric decomposition of powers of parameter ideals in a Noetherian local ring*, Tokyo J. Math., to appear. - William Heinzer, L. J. Ratliff Jr., and Kishor Shah,
*Parametric decomposition of monomial ideals. I*, Houston J. Math.**21**(1995), no. 1, 29–52. MR**1331242** - Paolo Valabrega and Giuseppe Valla,
*Form rings and regular sequences*, Nagoya Math. J.**72**(1978), 93–101. MR**514892**

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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