The depth of an ideal with a given Hilbert function

Abstract

Let $A=K[x_1,...,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $°x_i=1$. Let $I$ be a homogeneous ideal of $A$ with $I\ne A$ and $H_{A/I}$ the Hilbert function of the quotient algebra $A/I$. Given a numerical function $H:\mathbb{N}\to \mathbb{N}$ satisfying $H=H_{A/I}$ for some homogeneous ideal $I$ of $A$, we write ${\mathcal{A}}_H$ for the set of those integers $0\le r\le n$ such that there exists a homogeneous ideal $I$ of $A$ with $H_{A/I}=H$ and with $\depth A / I = r$. It will be proved that one has either ${\mathcal{A}}_H=\{0,1,...,b\}$ for some $0\le b\le n$ or $|{\mathcal{A}}_H|=1$.

6 pages, change the numbering of theorems