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.
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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
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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Cường, Đ.T., Nam, P.H. & Quý, P.H. On the Length Function of Saturations of Ideal Powers. Acta Math Vietnam 43, 275–288 (2018). https://doi.org/10.1007/s40306-018-0245-4
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DOI: https://doi.org/10.1007/s40306-018-0245-4
Keywords
- Saturation of ideal powers
- Hilbert polynomial of Artinian modules
- Rees polynomial
- Almost p-standard system of parameters