2018/08/09 by Fakhari, S. A. Seyed
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1808.03189
Let \mathbbK be a field and S=\mathbbK[x1,…,xn] be the polynomial ring in n variables over \mathbbK. Assume that G is a graph with edge ideal I(G). We prove that the modules S/I(G)k and I(G)k/I(G)k+1 satisfy Stanley's inequality for every integer k≫ 0. If G is a non-bipartite graph, we show that the ideals I(G)k satisfy Stanley's inequality for all k≫ 0. For every connected bipartite graph G (with at least one edge), we prove that \rm sdepth(I(G)k)≥ 2, for any positive integer k≤ \rm girth(G)/2+1. This result partially answers a question asked in [20]. For any proper monomial ideal I of S, it is shown that the sequence \\rm depth(Ik/Ik+1)\k=0∞ is convergent and limk→∞\rm depth(Ik/Ik+1)=n-ℓ(I), where ℓ(I) denotes the analytic spread of I. Furthermore, it is proved that for any monomial ideal I, there exists an integer s such that \rm depth (S/Ism) ≤ \rm depth (S/I),for every integer m≥ 1. We also determine a value s for which the above inequality holds. If I is an integrally closed ideal, we show that \rm depth(S/Im)≤ \rm depth(S/I), for every integer m≥ 1. As a consequence, we obtain that for any integrally closed monomial ideal I and any integer m≥ 1, we have \rm Ass(S/I)⊆ \rm Ass(S/Im). \endabstract