Journal of the Mathematical Society of Japan

On the sequential polynomial type of modules

Shiro GOTO and Le Thanh NHAN

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $M$ be a finitely generated module over a Noetherian local ring $R$. The sequential polynomial type $\mathrm{sp}(M)$ of $M$ was recently introduced by Nhan, Dung and Chau, which measures how far the module $M$ is from the class of sequentially Cohen–Macaulay modules. The present paper purposes to give a parametric characterization for $M$ to have $\mathrm{sp}(M)\le s$, where $s\ge -1$ is an integer. We also study the sequential polynomial type of certain specific rings and modules. As an application, we give an inequality between $\mathrm{sp}(S)$ and $\mathrm{sp}(S^G) $, where $S$ is a Noetherian local ring and $G$ is a finite subgroup of $\mathrm{Aut}S$ such that the order of $G$ is invertible in $S$.

Article information

J. Math. Soc. Japan, Volume 70, Number 1 (2018), 365-385.

First available in Project Euclid: 26 January 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13E05: Noetherian rings and modules 13C14: Cohen-Macaulay modules [See also 13H10] 13D45: Local cohomology [See also 14B15]

sequentially Cohen–Macaulay module strict $M$-sequence in dimension $>s$ distinguished system of parameters


GOTO, Shiro; NHAN, Le Thanh. On the sequential polynomial type of modules. J. Math. Soc. Japan 70 (2018), no. 1, 365--385. doi:10.2969/jmsj/07017535.

Export citation


  • J. W. Brewer and E. A. Rutter, Must $R$ be Noetherian if $R^G$ is Noetherian, \doihref10.1080/00927877708822205Comm. Algebra, 5 (1977), 969–979.
  • M. Brodmann and L. T. Nhan, On canonical Cohen–Macaulay modules, \doihref10.1016/j.jalgebra.2012.09.002J. Algebra, 371 (2012), 480–491.
  • M. Brodmann and L. T. Nhan, A finiteness result for associated primes of certain Ext-modules, \doihref10.1080/00927870701869543Comm. Algebra, 36 (2008), 1527–1536.
  • M. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge University Press, 1998.
  • N. T. Cuong, On the least degree of polynomials bounding above the differences between lengths and multiplicities of certain systems of parameters in local rings, \doihref10.1017/S0027763000003925Nagoya Math. J., 125 (1992), 105–114.
  • N. T. Cuong, p-standard systems of parameters and p-standard ideals in local rings, Acta Math. Vietnam., 20 (1995), 145–161.
  • N. T. Cuong and D. T. Cuong, Local cohomology annihilators and Macaulayfication, \doihref10.1007/s40306-016-0185-9Acta Math. Vietnam, 42 (2017), 37–60.
  • N. T. Cuong, D. T. Cuong and H. L. Truong, On a new invariant of finitely generated modules over local rings, \doihref10.1142/S0219498810004324J. Algebra Appl., 9 (2010), 959–976.
  • N. T. Cuong and L. T. Nhan, Pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay modules, \doihref10.1016/S0021-8693(03)00225-4J. Algebra, 267 (2003), 156–177.
  • S. Goto, On Buchsbaum ring, \doihref10.1016/0021-8693(80)90160-XJ. Algebra, 67 (1980), 272–279.
  • S. Goto, Y. Horiuchi and H. Sakurai, Sequentially Cohen-Macaulayness versus parametric decomposition of powers of parameter ideals, \doihref10.1216/JCA-2010-2-1-37J. Commut. Algebra, 2 (2010), 37–54.
  • T. Kawasaki, On arithmetic Macaulayfication of Noetherian rings, \doihref10.1090/S0002-9947-01-02817-3 Trans. Amer. Math. Soc., 354 (2002), 123–149.
  • I. G. Macdonald, Secondary representation of modules over a commutative ring, Symposia Mathematica, 11 (1973), 23–43.
  • L. T. Nhan, T. D. Dung and T. D. M. Chau, A measure of non-sequentially Cohen-Macaulayness of finitely generated modules, \doihref10.1016/j.jalgebra.2016.08.028J. Algebra, 468 (2016), 275–295.
  • L. T. Nhan and N. V. Hoang, A finiteness result for attached primes of Artinian local cohomology modules, \doihref10.1142/S0219498813500631J. Algebra Appl., 13 (2014), 1350063, 14pp.
  • L. T. Nhan, N. T. K. Nga and P. H. Khanh, Non-Cohen–Macaulay locus and non generalized Cohen–Macaulay locus, \doihref10.1080/00927872.2013.811675Comm. Algebra, 42 (2014), 4414–4425.
  • L. T. Nhan and P. H. Quy, Attached primes of local cohomology modules under localization and completion, \doihref10.1016/j.jalgebra.2014.08.043J. Algebra, 420 (2014), 475–485.
  • P. Schenzel, On the dimension filtration and Cohen–Macaulay filtered modules, Commutative algebra and algebraic geometry (Ferrara), 245–264, Lecture Notes in Pure and Appl. Math., 206, Dekker, New York, 1999.
  • \hskip-0.5ex N. Taniguchi, T. T. Phuong, N. T. Dung and T. N. An, Topics on sequentially Cohen–Macaulay modules, J. Commut. Algebra, to appear.