An Increasing normalized depth function

2023/09/25 by Fakhari, S. A. Seyed
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.13892

Abstract

Let \mathbbK be a field and S=\mathbbK[x₁,…,x_n] be the polynomial ring in n variables over \mathbbK. Assume that I is a squarefree monomial ideal of S. For every integer k≥ 1, we denote the k-th squarefree power of I by I^[k]. The normalized depth function of I is defined as g_I(k)=\rm depth(S/I^[k])-(d_k-1), where d_k denotes the minimum degree of monomials belonging to I^[k]. Erey, Herzog, Hibi and Saeedi Madani conjectured that for any squarefree monomial ideal I, the function g_I(k) is nonincreasing. In this short note, we provide a counterexample for this conjecture. Our example in fact shows that g_I(2)-g_I(1) can be arbitrarily large.

Related

• The normalized depth function of squarefree powers
• Behaviour of the normalized depth function
• Depth of factors of square free monomial ideals
• Upper bounds of depth of monomial ideals
• Depth of some special monomial ideals