On the depth and Stanley depth of the integral closure of powers of monomial ideals

摘要

Let K be a field and S = K[ x(1), ..., x(n)] be the polynomial ring in n variables over K. For any ;monomial ideal I, we denote its integral closure by (I) over bar. Assume that G is a graph with edge ideal I (G). We prove that the modules S/<(I(G)(k))over bar> and <(I(G)(k))over bar>/<(I(G)(k+1))over bar> satisfy Stanley's inequality for every integer k 0. If G is a non-bipartite graph, we show that the ideals <(I(G)(k))over bar> satisfy Stanley's inequality for all k 0. For every connected bipartite graph G (with at least one edge), we prove that sdepth(I (G)(k)) >= 2, for any positive integer k <= girth(G)/2 + 1. This result partially answers a question asked in Seyed Fakhari (J Algebra 489: 463-474, 2017). For any proper monomial ideal I of S, it is shown that the sequence {depth((I-k) over bar/(Ik+1) over bar)}(k=0)(infinity) is convergent and lim(k ->infinity) depth((I-k) over bar/(Ik+1) over bar) = 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