2022/10/01 by Ficarra, Antonino, Herzog, Jürgen, Hibi, Takayuki · 2 citations
#05C70 #05E40 #13C15 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2210.00210
Let I⊂ S=K[x1,…,xn] be a squarefree monomial ideal, K a field. The kth squarefree power I[k] of I is the monomial ideal of S generated by all squarefree monomials belonging to Ik. The biggest integer k such that I[k]≠(0) is called the monomial grade of I and it is denoted by ν(I). Let dk be the minimum degree of the monomials belonging to I[k]. Then, depth(S/I[k])≥ dk-1 for all 1≤ k≤ν(I). The normalized depth function of I is defined as gI(k)=depth(S/I[k])-(dk-1), 1≤ k≤ν(I). It is expected that gI(k) is a non-increasing function for any I. In this article we study the behaviour of gI(k) under various operations on monomial ideals. Our main result characterizes all cochordal graphs G such that for the edge ideal I(G) of G we have gI(G)(1)=0. They are precisely all cochordal graphs G whose complementary graph Gc is connected and has a cut vertex. As a far-reaching application, for given integers 1≤ s