## Parametric decomposition of powers of parameter ideals and sequentially Cohen-Macaulay modules

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- by Nguyen Tu Cuong and Hoang Le Truong
- Proc. Amer. Math. Soc.
**137**(2009), 19-26 - DOI: https://doi.org/10.1090/S0002-9939-08-09437-9
- Published electronically: July 29, 2008
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## Abstract:

Let $M$ be a finitely generated module of dimension $d$ over a Noetherian local ring $(R,\mathfrak {m})$ and $\mathfrak {q}$ an ideal generated by a system of parameters $\underline {x} = (x_1,\ldots , x_d)$ of $M$. For each positive integer $n$, set \[ \Lambda _{d,n}=\{ \alpha =(\alpha _1,\ldots ,\alpha _d)\in \mathbb {Z}^d|\alpha _i\geqslant 1, 1\leqslant i\leqslant d \text { and } \sum \limits _{i=1}^d\alpha _i=d+n-1\}\] and $\mathfrak {q}(\alpha )=(x_1^{\alpha _1},\ldots ,x_d^{\alpha _d})$ for each $\alpha \in \Lambda _{d,n}$. Then we prove in this note that $M$ is a sequentially Cohen-Macaulay module if and only if there exists a good system of parameters $\underline {x}$ such that the equality $\mathfrak {q}^nM=\bigcap \limits _{\alpha \in \Lambda _{d,n}}\mathfrak {q}(\alpha )M$ holds true for all $n\ge 1$. As an application, we show that the sequentially Cohen-Macaulayness of a module can be characterized by a very special expression of the Hilbert-Samuel polynomial of a good parameter ideal.## References

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## Bibliographic Information

**Nguyen Tu Cuong**- Affiliation: Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Viet Nam
- Email: ntcuong@math.ac.vn
**Hoang Le Truong**- Affiliation: Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Viet Nam
- MR Author ID: 842253
- Email: hltruong@math.ac.vn
- Received by editor(s): November 15, 2006
- Received by editor(s) in revised form: September 11, 2007, and November 18, 2007
- Published electronically: July 29, 2008
- Communicated by: Bernd Ulrich
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 19-26 - MSC (2000): Primary 13H10; Secondary 13H99
- DOI: https://doi.org/10.1090/S0002-9939-08-09437-9
- MathSciNet review: 2439420