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

[摘要] Let G be a graph with n vertices and let S = K[x(1),...,x(n)] be the polynomial ring in n variables over a field K. Assume that J(G) is the cover ideal of G and J(G)((k)) is its k-th symbolic power. We prove that the sequences {sdepth(S/J(G)((k)))}(k=1)(infinity), and {sdepth(S/J(G)((k)))}(k=1)(infinity) are non-increming and hence convergent. Suppose that nu(o)(G) denotes the ordered matching number of G. We show that for every integer k >= 2 nu(o) (G) - 1, the modules J(G)((k)) and S/J(G)((k)) satisfy the Stanley's inequality. We also provide an alternative proof for [9, Theorem 3.4] which states that depth(S/J(G)((k))) = n - nu(o)(G) - 1, for every integer k >= 2 nu(o)(G) - 1. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Stanley depth;Cover ideal;Symbolic power;Ordered matching number