Parametric decomposition of powers of parameter ideals and sequentially Cohen-Macaulay modules
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- by Nguyen Tu Cuong and Hoang Le Truong PDF
- Proc. Amer. Math. Soc. 137 (2009), 19-26 Request permission
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
- N. T. Cuong and D. T. Cuong, On sequentially Cohen-Macaulay modules, Kodai Math. J., 30 (2007), 409-428.
- Nguyen Tu Cuong and Le Thanh Nhan, Pseudo Cohen-Macaulay and pseudo generalized Cohen-Macaulay modules, J. Algebra 267 (2003), no. 1, 156–177. MR 1993472, DOI 10.1016/S0021-8693(03)00225-4
- Shiro Goto, Approximately Cohen-Macaulay rings, J. Algebra 76 (1982), no. 1, 214–225. MR 659220, DOI 10.1016/0021-8693(82)90248-4
- Shiro Goto and Yasuhiro Shimoda, Parametric decomposition of powers of ideals versus regularity of sequences, Proc. Amer. Math. Soc. 132 (2004), no. 4, 929–933. MR 2045406, DOI 10.1090/S0002-9939-03-07160-0
- Shiro Goto and Yasuhiro Shimoda, On the parametric decomposition of powers of parameter ideals in a Noetherian local ring, Tokyo J. Math. 27 (2004), no. 1, 125–135. MR 2060079, DOI 10.3836/tjm/1244208479
- 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
- Peter Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, Commutative algebra and algebraic geometry (Ferrara), Lecture Notes in Pure and Appl. Math., vol. 206, Dekker, New York, 1999, pp. 245–264. MR 1702109, DOI 10.1090/conm/239/03606
- Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
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
- 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