Stanley depth of the integral closure of monomial ideals

• Autores: S. A. Seyed Fakhari
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 64, Fasc. 3, 2013, págs. 351-362
• Idioma: inglés
• DOI: 10.1007/s13348-012-0077-9
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• Resumen

  • Let I be a monomial ideal in the polynomial ring S=\mathbb K [x_1,\dots ,x_n]. We study the Stanley depth of the integral closure \overline{I} of I. We prove that for every integer k\ge 1, the inequalities \text{ sdepth} (S/\overline{I^k}) \le \text{ sdepth} (S/\overline{I}) and \text{ sdepth} (\overline{I^k}) \le \text{ sdepth} (\overline{I}) hold. We also prove that for every monomial ideal I\subset S there exist integers k_1,k_2\ge 1, such that for every s\ge 1, the inequalities \text{ sdepth} (S/I^{sk_1}) \le \text{ sdepth} (S/\overline{I}) and \text{ sdepth} (I^{sk_2}) \le \text{ sdepth} (\overline{I}) hold. In particular, \min _k \{\text{ sdepth} (S/I^k)\} \le \text{ sdepth} (S/\overline{I}) and \min _k \{\text{ sdepth} (I^k)\} \le \text{ sdepth} (\overline{I}). We conjecture that for every integrally closed monomial ideal I, the inequalities \text{ sdepth}(S/I)\ge n-\ell (I) and \text{ sdepth} (I)\ge n-\ell (I)+1 hold, where \ell (I) is the analytic spread of I. Assuming the conjecture is true, it follows together with the Burch’s inequality that Stanley’s conjecture holds for I^k and S/I^k for k\gg 0, provided that I is a normal ideal.