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Depth functions of symbolic powers of homogeneous ideals

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 \(t \ge 1\).

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Notes

  1. 1.

    The notation of \(P_F\) in [16] is different.

  2. 2.

    There is a typo in [37, Lemma 1.5]. In the formula for \(M_1\) one has to replace \(\text {Assm}(R)\setminus {\mathcal V}(I_F)\) by \(\text {Assm}(R)\cap {\mathcal V}(I_F)\).

  3. 3.

    The proof for Proposition 3.2(ii) and (iii) of [30] has errors, which can be corrected as follows. For Proposition 3.2(ii), we consider the exact consequence \(0 \rightarrow M \cap N \rightarrow L \rightarrow (L/M) \oplus (L/N)\) and apply Corollary 2.5(i) and Lemma 3.1. Proposition 3.2(iii) follows from the exact sequence \(M \oplus N \rightarrow L \rightarrow L/(M+N) \rightarrow 0\) by applying Corollary 2.5(ii) and Lemma 3.1.

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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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Correspondence to Ngo Viet Trung.

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Nguyen, H.D., Trung, N.V. Depth functions of symbolic powers of homogeneous ideals. Invent. math. 218, 779–827 (2019). https://doi.org/10.1007/s00222-019-00897-y

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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