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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Nguyen Tu Cuong and Hoang Le Truong
Journal: Proc. Amer. Math. Soc. 137 (2009), 19-26
MSC (2000): Primary 13H10; Secondary 13H99
Published electronically: July 29, 2008
MathSciNet review: 2439420
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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

$\displaystyle \Lambda_{d,n}=\{ \alpha =(\alpha_1,\ldots,\alpha_d)\in\mathbb{Z}^d\vert\alpha_i\geqslant 1, 1\leqslant i\leqslant d$    and $\displaystyle \sum\limits_{i=1}^d\alpha_i=d+n-1\}$

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\ge1$. 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.


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Additional 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
Email: hltruong@math.ac.vn

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09437-9
PII: S 0002-9939(08)09437-9
Keywords: Parametric decomposition, sequentially Cohen-Macaulay module, dimension filtration, good system of parameters.
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.