Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$. 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 $0 \leq r \leq n$ 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 $0 \leq b \leq n$ or $|\mathcal{ A}_H| = 1$.
Categories: math.AC