AN INCREASING NORMALIZED DEPTH FUNCTION

Let 𝕂 be a field and S=𝕂[x1,…⁡,xn] be the polynomial ring in n variables over 𝕂. 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 gI(k)=depth⁡(S∕I[k])−(dk−1), where dk 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 gI(k) is nonincreasing. In this short note, we provide a counterexample for this conjecture. Our example in fact shows that gI(2)−gI(1) can be arbitrarily large.