Remarks on the Hilbert depth of squarefree monomial ideals
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911853535297536 |
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| author | Balanescu, Silviu Cimpoeas, Mircea |
| author_facet | Balanescu, Silviu Cimpoeas, Mircea |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_12339 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Remarks on the Hilbert depth of squarefree monomial ideals Balanescu, Silviu Cimpoeas, Mircea Commutative Algebra 05A18, 06A07, 13C15, 13P10, 13F20 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. |
| title | Remarks on the Hilbert depth of squarefree monomial ideals |
| topic | Commutative Algebra 05A18, 06A07, 13C15, 13P10, 13F20 |
| url | https://arxiv.org/abs/2310.12339 |