On the Length Function of Saturations of Ideal Powers
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On the Length Function of Saturations of Ideal Powers Đoàn Trung Cường1,2 Phạm Hùng Quý4,5
` Nam3 · · Phạm Hông
Received: 29 December 2016 / Revised: 10 August 2017 / Accepted: 21 October 2017 / Published online: 2 February 2018 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018
Abstract For an ideal I in a Noetherian local ring (R, m), we prove that the integer-valued 0 (R/I n+1 )) is a polynomial for n big enough if either I is a principal ideal function R (Hm 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 0 (R/I n+1 )) is not a polynomial for n 0. that the function R (Hm Keywords Saturation of ideal powers · Hilbert polynomial of Artinian modules · Rees polynomial · Almost p-standard system of parameters
Đoàn Trung Cường
[email protected] Phạm Hô` ng Nam [email protected] Phạm Hùng Quý [email protected] 1
Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam
2
The Graduate University of Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam
3
University of Sciences, Thai Nguyen University, Thai Nguyen, Vietnam
4
Department of Mathematics, FPT University, Hanoi, Vietnam
5
Thang Long Institute of Mathematics and Applied Sciences, Thang Long University, Hanoi, Vietnam
276
D. T. Cuong et al.
Mathematics Subject Classification (2010) 13H15 · 13D40 · 13D45
1 Introduction Let (R, m) be a Noetherian local ring and let I ⊂ R be an ideal. There is a numerical function attached to I , 0 (R/I n+1 )). h0I : Z≥0 → Z≥0 , n → R (Hm
If I is m-primary then h0I (n) = (R/I n+1 ) is the Hilbert-Samuel function. So it is of polynomial type, that means, there is a polynomial HI (n) such that h0I (n) = HI (n) for all n 0. For a general ideal I , it is natural to ask whether the function h0I (n) is of polynomial type. Unfortunately, it is not always the case. In [18], Ulrich and Validashti proved that the limit h0 (n) lim sup d! I d n n is a finite number. They showed that this limit can be considered as a generalization of the Buchsbaum-Rim multiplicity and called it the -multiplicity. This numerical invariant -multiplicity is denoted by (I ) and is very useful in the theory of equisingularity (see [14]). Nevertheless, Cutkosky, Ha, Srinivasan, and Theodorescu [9, Theorem 2.2] gave an interesting example of a four-dimensional local ring R and an ideal I such that (I ) = lim 4! n→∞
h0I (n) , n4
is an irrational number. Thus, in this example, h0I (n) is not of polynomial type, and the answer to the previous question
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