On the index of depth stability of symbolic powers of cover ideals of graphs
type: Item
instance of: scholarly article
title: On the index of depth stability of symbolic powers of cover ideals of graphs
author: S. A. Seyed Fakhari
author: Siamak Yassemi
published in: Acta Mathematica Vietnamica
publication date: 2024-12-09T00:00:00Z
review text: Let \(G\) be a graph with \(n\) vertices and \(S=K[x_{1},\dots,x_n]\) be the polynomial ring in \(n\) variables over a field \(K\). The paper provides a combinatorial upper bound for the index of depth stability of symbolic powers of \(J(G)\), the cover ideal of the graph \(G\). The index of depth stability of powers of an ideal \(I\) of \(S\) here is the smallest integer \(k\geq 1\) such that \(\operatorname{depth}(S/I^m)=\lim_{t\to \infty}\operatorname{depth}(S/I^t)\) for all \(m\geq k\). This is therefore helpful for determining the limit of the sequence \(\{\operatorname{depth}(S/I^k)\}_{k\in\mathbb{N}}\) which is itself of great interest. Also, same problems are considered and explained for symbolic powers \(I^{(k)}\) of an ideal and the paper computes the depth of symbolic powers of cover ideals of fully clique-whiskered graphs. It is already known ([\textit{L. T. Hoa} et al., J. Algebra 473, 307--323 (2017; Zbl 1358.13020)]) that \(\operatorname{dstab}(J(G)\leq 2\operatorname{ord-match}(G)-1\) and the present paper provides an improvement of this bound (Theorem 3.1). Here \(\operatorname{ord-match}(G)\) denotes the order matching of a graph \(G\) (Definition 2.2). The authors also determine a class of graphs \(G\) with the property that the Castelnuovo-Mumford regularity of \(S/I(G)\) is equal to the induced matching number of \(G\) (another invariant introduced in Section 2, page 368 of the paper). Recall that if \(M\) is a finitely generated graded \(S\)-Module, then Castelnuovo-Mumford regularity (or simply, regularity) of M is defined as \(\max\{j-i\mid\mathrm{Tor}_i(K,M)_j\neq 0\}\).
reviewed by: Mohajer Abolfazl
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 05E40
Mathematics Subject Classification ID: 13C13
zbMATH DE Number: 7955883
zbMATH Keywords: Castelnuovo-Mumford regularity
zbMATH Keywords: cover ideal
zbMATH Keywords: depth
zbMATH Keywords: symbolic power
zbMATH Keywords: ordered matching number
MaRDI profile type: Publication
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DOI: 10.1007/S40306-024-00550-8
description: scientific article; zbMATH DE number 7955883