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Depth functions of symbolic powers of homogeneous ideals

  • Hop Dang Nguyen
  • Ngo Viet TrungEmail author
Article

Abstract

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function \({{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R -{{\,\mathrm{pd}\,}}I^{(t)} - 1\), where \(I^{(t)}\) denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and \({{\,\mathrm{pd}\,}}\) denotes the projective dimension. It has been an open question whether the function \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is non-increasing if I is a squarefree monomial ideal. We show that \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is almost non-increasing in the sense that \({{\,\mathrm{depth}\,}}R/I^{(s)} \ge {{\,\mathrm{depth}\,}}R/I^{(t)}\) for all \(s \ge 1\) and \(t \in E(s)\), where
$$\begin{aligned} E(s) = \bigcup _{i \ge 1}\{t \in {\mathbb {N}}|\ i(s-1)+1 \le t \le is\} \end{aligned}$$
(which contains all integers \(t \ge (s-1)^2+1\)). The range E(s) is the best possible since we can find squarefree monomial ideals I such that \({{\,\mathrm{depth}\,}}R/I^{(s)} < {{\,\mathrm{depth}\,}}R/I^{(t)}\) for \(t \not \in E(s)\), which gives a negative answer to the above question. Another open question asks whether the function \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is always constant for \(t \gg 0\). We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that \(I^{(t)}\) is integrally closed for \(t \gg 0\) (e.g. if I is a squarefree monomial ideal), then \({{\,\mathrm{depth}\,}}R/I^{(t)}\) is constant for \(t \gg 0\) with
$$\begin{aligned} \lim _{t \rightarrow \infty }{{\,\mathrm{depth}\,}}R/I^{(t)} = \dim R - \dim \oplus _{t \ge 0}I^{(t)}/{\mathfrak {m}}I^{(t)}. \end{aligned}$$
Our last result (which is the main contribution of this paper) shows that for any positive numerical function \(\phi (t)\) which is periodic for \(t \gg 0\), there exist a polynomial ring R and a homogeneous ideal I such that \({{\,\mathrm{depth}\,}}R/I^{(t)} = \phi (t)\) for all \(t \ge 1\). As a consequence, for any non-negative numerical function \(\psi (t)\) which is periodic for \(t \gg 0\), there is a homogeneous ideal I and a number c such that \({{\,\mathrm{pd}\,}}I^{(t)} = \psi (t) + c\) for all \(t \ge 1\).

Keywords

Symbolic power Projective dimension Depth Asymptotic behavior Monomial ideal Integrally closed ideal Degree complex Local cohomology Bertini-type theorem System of linear diophantine inequalities 

Mathematics Subject Classification

13C15 14B05 

Notes

Acknowledgements

Hop Dang Nguyen is partially supported by Project CT 0000.03/19-21 of Vietnam Academy of Science and Technology. Ngo Viet Trung is partially supported by Vietnam National Foundation for Science and Technology Development. Part of this work was done during research stays of the authors at Vietnam Institute for Advanced Study in Mathematics. The authors would like to thank Huy Tài Hà and Tran Nam Trung for their collaboration on the joint paper [11] which initiated this work. They are also grateful to the referee for many suggestions which help improve the presentation of the paper. After the revision of this paper, the authors have been informed that Theorem 2.7 has been recently obtained in a modified form by different methods by J. Montano and L. Nunez-Betancourt (arXiv:1809.02308) and S. A. Seyed Fakhari (arXiv:1812.03742).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.International Centre for Research and Postgraduate Training, Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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