Stability of depths of powers of edge ideals

Abstract

Abstract. Let G be a graph and let I := I(G) be its edge ideal. In this paper, we provide an upper bound of n from which depth R/I (G) n is stationary, and compute this limit explicitly. This bound is always achieved if G has no cycles of length 4 and every its connected component is either a tree or a unicyclic graph. IntroductionLet R = K[x 1 , . . . , x r ] be a polynomial ring over a field K and I a homogeneous ideal in R. Brodmann [2] showed that depth R/I n is a constant for sufficiently large n. Moreover limwhere ℓ(I) is the analytic spread of I. It was shown in [6, Proposition 3.3] that this is an equality when the associated graded ring of I is Cohen-Macaulay. We call the smallest number n 0 such that depth R/I n = depth R/I n 0 for all n n 0 , the index of depth stability of I, and denote this number by dstab(I). It is of natural interest to find a bound for dstab(I). As until now we only know effective bounds of dstab(I) for few special classes of ideals I, such as complete intersection ideals (see [5]), square-free Veronese ideals (see [8]), polymatroidal ideals (see [10]). In this paper we will study this problem for edge ideals.From now on, every graph G is assumed to be simple (i.e., a finite, undirected, loopless and without multiple edges) without isolated vertices on the vertex set V (G) = [r] := {1, . . . , r} and the edge set E(G) unless otherwise indicated. We associate to G the quadratic squarefree monomial idealwhich is called the edge ideal of G.If I is a polymatroidal ideal in R, Herzog and Qureshi proved that dstab(I) < dim R and they asked whether dstab(I) < dim R for all Stanley-Reisner ideals I in R (see [10]). For a graph G, if every its connected component is nonbipartite, then we can see that dstab(I(G)) < dim R from [4]. In general, there is not an absolute bound of dstab(I(G)) even in the case G is a tree (see [20]). In this paper we will establish a bound of dstab(I(G)) for any graph G. In particular, dstab(I(G)) < dim R.1991 Mathematics Subject Classification. 13D45, 05C90, 05E40, 05E45.

Saturation and associated primes of powers of edge ideals

“…[1,2,4,5,7,10,13,17]), but little is known about a particular power I t (except for its Cohen-Macaulay property [12,15]). There was a combinatorial description of the associated primes of every power of the cover ideals of graphs by Francisco, Ha and Van Tuyl [3].…”

The behavior of depth functions of cover ideals of unimodular hypergraphs

“…Effective bounds of dstab(I) are known only for a few special classes of ideals I, such as complete intersection ideals (see [5]), squarefree Veronese ideals (see [10]), polymatroidal ideals (see [13]), edge ideals (see [25]). For any cover ideal of a unimodular hypergraph H, we establish the universal and effective bound for dstab(J(H)).…”

Stability of depth and Stanley depth of symbolic powers of squarefree monomial ideals

“…Furthermore, they asked whether it is true that for every squarefree monomial ideal I, the inequality dstab(I) < n holds. Trung [21] investigated the case of edge ideals and proved that for any edge ideal I(G) ⊂ S, the limit lim k→∞ depth(S/I(G) k ) is n − ℓ(I(G)) which is equal to the number of bipartite connected components of G. Moreover, in the same paper, it is shown that for any graph G with n vertices, we have dstab(I(G)) < n. This gives a positive answer to the above mentioned question of Herzog and Qureshi, in the case of edge ideals. It is also of interest to consider similar problems for the symbolic powers of monomial ideals.…”