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

2017/09/10 by Fakhari, S. A. Seyed
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1709.03882

Abstract

Let G be a graph with n vertices and let S=\mathbbK[x₁,…,x_n] be the polynomial ring in n variables over a field \mathbbK. 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 \\rm sdepth(S/J(G)^(k))\_k=1^∞ and \\rm sdepth(J(G)^(k))\_k=1^∞ are non-increasing and hence convergent. Suppose that ν_o(G) denotes the ordered matching number of G. We show that for every integer k≥ 2ν_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 \cite[Theorem 3.4]hktt which states that \rm depth(S/J(G)^(k))=n-ν_o(G)-1, for every integer k≥ 2ν_o(G)-1.

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