Remarks on the Hilbert depth of squarefree monomial ideals

Silviu Balanescu; Mircea Cimpoeas

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arXiv · 2310.12339

Remarks on the Hilbert depth of squarefree monomial ideals

Silviu Balanescu·Mircea Cimpoeas

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Abstract

Let $K$ be a infinite field, $S=K[x_1,\ldots,x_n]$ and $0\subset I\subsetneq J\subset S$ two squarefree monomial ideals. In a previous paper we proved a new formula for the Hilbert depth of $J/I$. In this paper, we illustrate how one can use the Stanley-Reisner correspondence between (relative) simplicial complexes and (quotients of) squarefree monomial ideals, in order to reobtain some basic properties of the Hilbert depth. More precisely, we show that $\operatorname{depth}(J/I)\leq \operatorname{hdepth}(J/I)\leq \dim(J/I)$. Also, we show that $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)+1$, if $S/I$ is Cohen-Macaulay.

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Silviu Balanescu, Mircea Cimpoeas. 2023-10-18. Remarks on the Hilbert depth of squarefree monomial ideals. https://arxiv.org/abs/2310.12339

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