Cohen–Macaulayness and Limit Behavior of Depth for Powers of Cover Ideals

Abstract

Let 𝕂 be a field, and let R = 𝕂[x 1,…, x n ] be the polynomial ring over 𝕂 in n indeterminates x 1,…, x n . Let G be a graph with vertex-set {x 1,…, x n }, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J (k) and J [k], respectively. In this paper, we give necessary and sufficient conditions for R/J k , R/J (k), and R/J [k] to be Cohen–Macaulay. We also study the limit behavior of the depths of these rings.