The depth of an ideal with a given Hilbert function

Details The depth of an ideal with a given Hilbert function The depth of an ideal with a given Hilbert function

2006-08-08

arxiv.org/abs/math/0608188v2

Proc. Amer. Math. Soc. 136 (2008), 1533--1538

Mathematics

Commutative Algebra

6 pages, change the numbering of theorems

Scientific paper

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 $|\mathcal{A}_H| = 1$.

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