Stability of Depth and Cohen-Macaulayness of Integral Closures of Powers of Monomial Ideals

Abstract

Let I be a monomial ideal in a polynomial ring \(R = k[x_{1},\dots ,x_{r}]\). In this paper, we give an upper bound on \(\overline {\text {dstab}} (I)\) in terms of r and the maximal generating degree d(I) of I such that \(\text {depth} R/\overline {I^{n}}\) is constant for all \(n\geqslant \overline {\text {dstab}}(I)\). As an application, we classify the class of monomial ideals I such that \(\overline {I^{n}}\) is Cohen-Macaulay for some integer n ≫ 0.

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Acknowledgements

This work is partially supported by NAFOSTED (Vietnam) under the grant number 101.04-2015.02.

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Correspondence to Le Tuan Hoa.

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Hoa, L.T., Trung, T.N. Stability of Depth and Cohen-Macaulayness of Integral Closures of Powers of Monomial Ideals. Acta Math Vietnam 43, 67–81 (2018). https://doi.org/10.1007/s40306-017-0225-0

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Keywords

  • Depth
  • Monomial ideal
  • Simplicial complex
  • Integral closure

Mathematics Subject Classification (2010)

  • 13D45
  • 05C90