Sharif Digital Repository

Let K be a field and S = K[x1,...,xn] be the polynomial ring in n variables over the field K. Let G be a forest with p connected components G1,...,Gp and let I = I(G) be its edge ideal in S. Suppose that di is the diameter of Gi, 1 ≤ i ≤ p, and consider d = max{di I 1 ≤ i ≤ p}. Morey has shown that for every t ≥ 1, the quantity max is a lower bound for depth(S/It). In this paper, we show that for every t ≥ 1, the mentioned quantity is also a lower bound for sdepth(S/It). By combining this inequality with Burch's inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem  

Let K be a field and S = K[x1,...,xn] be the polynomial ring in n variables over the field K. In this paper, it is shown that Stanley's conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree  

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...