Depth and Stanley depth of symbolic powers of cover ideals of graphs

Let K be a field and S = K[x 1 , . . . , x n ] be the polynomial ring in n variables over K. 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 sdepth(I(G) k ) ≥ 2, for any positive integer k ≤ 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 {depth(I k /I k+1 )} ∞ k=0 is convergent and lim k→∞ depth(I k /I k+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 depth(S/I sm ) ≤ 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 depth(S/I m ) ≤ 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 Ass(S/I) ⊆ Ass(S/I m ).

show abstract

Depth and Stanley depth of symbolic powers of cover ideals of graphs

Cited by 9 publications

References 21 publications

Homological and Combinatorial Properties of Powers of Cover Ideals of Graphs

Homological and Combinatorial Properties of Powers of Cover Ideals of Graphs

Regularity of symbolic powers of cover ideals of graphs

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

Contact Info

Product

Resources

About