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arXiv · 2501.07829

Restrictions on Hilbert coefficients give depths of graded domains

Cheng Meng
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Abstract

In this paper, we prove that if $P$ is a homogeneous prime ideal inside a standard graded polynomial ring $S$ with $\dim(S/P)=d$, and for $s \leq d$, adjoining $s$ general linear forms to the prime ideal changes the $(d-s)$-th Hilbert coefficient by 1, then $\text{depth}(S/P)=s-1$. This criterion also tells us about possible restrictions on the generic initial ideal of a prime ideal inside a polynomial ring.

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Cheng Meng. 2025-01-14. Restrictions on Hilbert coefficients give depths of graded domains. https://arxiv.org/abs/2501.07829

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