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Acta Mathematica Vietnamica

, Volume 43, Issue 2, pp 275–288 | Cite as

On the Length Function of Saturations of Ideal Powers

  • Đoàn Trung Cường
  • Phạm Hồng Nam
  • Phạm Hùng Quý
Article

Abstract

For an ideal I in a Noetherian local ring \((R, \mathfrak {m})\), we prove that the integer-valued function \(\ell _{R}(H^{0}_{\mathfrak {m}}(R/I^{n + 1}))\) is a polynomial for n big enough if either I is a principal ideal or I is generated by part of an almost p-standard system of parameters and R is unmixed. Furthermore, we are able to compute the coefficients of this polynomial in terms of length of certain local cohomology modules and usual multiplicity if either the ideal is principal or it is generated by part of a standard system of parameters in a generalized Cohen-Macaulay ring. We also give an example of an ideal generated by part of a system of parameters such that the function \(\ell _{R}(H^{0}_{\mathfrak {m}} (R/I^{n + 1}))\) is not a polynomial for n ≫ 0.

Keywords

Saturation of ideal powers Hilbert polynomial of Artinian modules Rees polynomial Almost p-standard system of parameters 

Mathematics Subject Classification (2010)

13H15 13D40 13D45 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions to improve the presentation of the paper. This paper was written while the third author was visiting Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for hospitality and financial support.

Funding Information

The first and the second authors are funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2015.26.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.The Graduate University of Science and TechnologyVietnam Academy of Science and TechnologyHanoiVietnam
  3. 3.University of SciencesThai Nguyen UniversityThai NguyenVietnam
  4. 4.Department of MathematicsFPT UniversityHanoiVietnam
  5. 5.Thang Long Institute of Mathematics and Applied SciencesThang Long UniversityHanoiVietnam

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