Nonincreasing depth functions of monomial ideals

Abstract

Given a nonincreasing function $f:{\mathbb{Z}}_{\ge 0}\setminus \{0\}\to {\mathbb{Z}}_{\ge 0}$ such that (i) $f(k)-f(k+1)\le 1$ for all $k\ge 1$ and (ii) if $a=f(1)$ and $b={\lim }_{k\to \infty }f(k)$, then $|f^{-1}(a)|\le |f^{-1}(a-1)|\le \cdots \le |f^{-1}(b+1)|$, a system of generators of a monomial ideal $I\subset K[x_1,\dots ,x_n]$ for which $\mathrm{depth}S/I^k=f(k)$ for all $k\ge 1$ is explicitly described. Furthermore, we give a characterization of triplets of integers $(n,d,r)$ with $n>0$, $d\ge 0$ and $r>0$ with the properties that there exists a monomial ideal $I\subset S=K[x_1,\dots ,x_n]$ for which ${\lim }_{k\to \infty }\mathrm{depth}S/I^k=d$ and $\mathrm{dstab}(I)=r$, where $\mathrm{dstab}(I)$ is the smallest integer $k_0\ge 1$ with $\mathrm{depth}S/I^{k_0}=\mathrm{depth}S/I^{k_0+1}=\mathrm{depth}S/I^{k_0+2}=\cdots$.

7 pages