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Filter regular sequence under small perturbations

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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. 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. 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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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Linquan Ma

  2. Department of Mathematics, FPT University, Hanoi, Vietnam

    Pham Hung Quy

  3. Department of Mathematics, Stockholm University, 10691, Stockholm, Sweden

    Ilya Smirnov

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  1. Linquan Ma
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  2. Pham Hung Quy
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Correspondence to Pham Hung Quy.

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Communicated by Vasudevan Srinivas.

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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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