Filter regular sequence under small perturbations

Abstract

We answer affirmatively a question of Srinivas–Trivedi (J Algebra 186(1):1–19, 1996): in a Noetherian local ring \((R,{{\,\mathrm{\mathfrak {m}}\,}})\), if \(f_1,\dots ,f_r\) is a filter-regular sequence and J is an ideal such that \((f_1, \ldots , f_r)+J\) is \({{\,\mathrm{\mathfrak {m}}\,}}\)-primary, then there exists \(N>0\) such that for any \(\varepsilon _1,\dots ,\varepsilon _r \in {{\,\mathrm{\mathfrak {m}}\,}}^N\), we have an equality of Hilbert functions: \(H(J, R/(f_1,\dots ,f_r))(n)=H(J, R/(f_1+\varepsilon _1,\dots , f_r+\varepsilon _r))(n)\) for all \(n\ge 0\). We also prove that the dimension of the non Cohen–Macaulay locus does not increase under small perturbations, generalizing another result of [20].

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Notes

  1. 1.

    Here \({{\,\mathrm{H}\,}}_{G_+}^i(G)\) denotes the i-th local cohomology module of G supported at the irrelevant ideal \(G_+=\oplus _{i>0}G_i\). It follows from the Čech complex characterization of local cohomology (via a homogeneous set of generators of \(G_+\)) that each \({{\,\mathrm{H}\,}}_{G_+}^i(G)\) is \(\mathbb {Z}\)-graded.

  2. 2.

    It should be noted that this inclusion is true when f is a parameter element [3, Remark 2.2, Lemma 3.7]

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Acknowledgements

This work has been done during a visit of the second and third authors to Purdue University in June 2019. The first author is supported in part by NSF Grant DMS \(\#1901672\) and by NSF Grant DMS \(\#1836867/1600198\) when preparing this paper. The second author is supported by a fund of Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2020.10. The third author’s stay at the Purdue University was supported by Stiftelsen G S Magnusons fond of Kungliga Vetenskapsakademien. The authors thank the referee for her/his suggestions that led to improvement of this paper.

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Ma, L., Quy, P.H. & Smirnov, I. Filter regular sequence under small perturbations. Math. Ann. 378, 243–254 (2020). https://doi.org/10.1007/s00208-020-02014-4

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