Parametric decomposition of powers of ideals versus regularity of sequences
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- by Shiro Goto and Yasuhiro Shimoda PDF
- Proc. Amer. Math. Soc. 132 (2004), 929-933 Request permission
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 $\Lambda _{s,n} = \{(\alpha _{1}, \alpha _{2}, \cdots , \alpha _{s}) \in {\mathbb {Z}}^{s} \mid \alpha _{i} \geq 1 \ \text {for all} \ 1 \leq i \leq s \ \text {and} \ \sum _{i=1}^{s}\alpha _{i} = s + n - 1\}$.References
- 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, DOI 10.1017/S0027763000018225
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
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 929-933
- MSC (2000): Primary 13H99
- DOI: https://doi.org/10.1090/S0002-9939-03-07160-0
- MathSciNet review: 2045406