Behaviour of the normalized depth function (Q6162139)

Summary: Let \(I\subset S=K[x_1,\ldots, x_n]\) be a squarefree monomial ideal, \(K\) a field. The \(k\)th squarefree power \(I^{[k]}\) of \(I\) is the monomial ideal of \(S\) generated by all squarefree monomials belonging to \(I^k\). The biggest integer \(k\) such that \(I^{[k]}\ne(0)\) is called the monomial grade of \(I\) and it is denoted by \(\nu(I)\). Let \(d_k\) be the minimum degree of the monomials belonging to \(I^{[k]}\). Then, \(\text{depth}(S/I^{[k]})\geqslant d_k-1\) for all \(1\leqslant k\leqslant\nu(I)\). The normalized depth function of \(I\) is defined as \(g_I(k)=\text{depth}(S/I^{[k]})-(d_k-1), 1\leqslant k\leqslant\nu(I)\). It is expected that \(g_I(k)\) is a non-increasing function for any \(I\). In this article we study the behaviour of \(g_I(k)\) 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 \(g_{I(G)}(1)=0\). They are precisely all cochordal graphs \(G\) whose complementary graph \(G^c\) is connected and has a cut vertex. As a far-reaching application, for given integers \(1\leqslant s<m\) we construct a graph \(G\) such that \(\nu(I(G))=m\) and \(g_{I(G)}(k)=0\) if and only if \(k=s+1,\ldots, m\). Finally, we show that any non-increasing function of non-negative integers is the normalized depth function of some squarefree monomial ideal.