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Stanley depth of the integral closure of monomial ideals

Seyed Fakhari, S. A ; Sharif University of Technology | 2013

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  1. Type of Document: Article
  2. DOI: 10.1007/s13348-012-0077-9
  3. Publisher: 2013
  4. Abstract:
  5. Let I be a monomial ideal in the polynomial ring S=K [x1,... xn]. We study the Stanley depth of the integral closure Ī of I. We prove that for every integer k ≥ 1, the inequalities (S/Ik) ≤ sdepth (S/Ī) and sdepth(Ik) ≤ sdepth(Ī) hold. We also prove that for every monomial ideal I⊂ S there exist integers k1,k2≥ 1, such that for every s≥ 1, the inequalities sdepth (S/Isk1) ≤ sdepth(S/Ī) and sdepth (Isk2) ≤ sdepth (Ī) hold. In particular, mink{sdepth(S/Ik)} ≤ sdepth(S/Ī) and min̄k {sdepth (Ik)}≤ sdepth(Ī). We conjecture that for every integrally closed monomial ideal I, the inequalities sdepth(S/I)≥ n-l (I) and sdepth (I)≥ n-l (I)+1 hold, where l (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 Ik and S/Ik for k ⋙ 0, provided that I is a normal ideal
  6. Keywords:
  7. Integral closure ; Integrally closed ideal ; Monomial ideal ; Normal ideal ; Stanley conjecture ; Stanley depth
  8. Source: Collectanea Mathematica ; Volume 64, Issue 3 , November , 2013 , Pages 351-362 ; 00100757 (ISSN)
  9. URL: http://link.springer.com/article/10.1007%2Fs13348-012-0077-9