The normalized depth function of squarefree powers

Abstract

The depth of squarefree powers of a squarefree monomial ideal is introduced. Let $I$ be a squarefree monomial ideal of the polynomial ring $S=K[x_1,\dots ,x_n]$. The $k$-th squarefree power $I^{[k]}$ of $I$ is the ideal of $S$ generated by those squarefree monomials $u_1\cdots u_k$ with each $u_i\in G(I)$, where $G(I)$ is the unique minimal system of monomial generators of $I$. Let $d_k$ denote the minimum degree of monomials belonging to $G(I^{[k]})$. One has $\mathrm{depth}(S/I^{[k]})\ge d_k-1$. Setting $g_I(k)=\mathrm{depth}(S/I^{[k]})-(d_k-1)$, one calls $g_I(k)$ the normalized depth function of $I$. The computational experience strongly invites us to propose the conjecture that the normalized depth function is nonincreasing. In the present paper, especially the normalized depth function of the edge ideal of a finite simple graph is deeply studied.