The depth of an ideal with a given Hilbert function

Abstract:

Let $A = K[x_1, \ldots , x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\operatorname {deg} x_i = 1$. Let $I$ be a homogeneous ideal of $A$ with $I \neq A$ 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 $\operatorname {depth} A / I = r$. It will be proved that one has either $\mathcal {A}_H = \{ 0, 1, \ldots , b \}$ for some $0 \leq b \leq n$ or $|{\mathcal {A}}_H| = 1$.