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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.18396 |
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| _version_ | 1866910100668547072 |
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| author | Das, Kanoy Kumar Roy, Amit Saha, Kamalesh |
| author_facet | Das, Kanoy Kumar Roy, Amit Saha, Kamalesh |
| contents | Let $I(Δ)^{[k]}$ denote the $k^{\text{th}}$ square-free power of the facet ideal of a simplicial complex $Δ$ in a polynomial ring $R$. Square-free powers are intimately related to the `Matching Theory' and `Ordinary Powers'. In this article, we show that if $Δ$ is a Cohen-Macaulay simplicial forest, then $R/I(Δ)^{[k]}$ is Cohen-Macaulay for all $k\ge 1$. This result is quite interesting since all ordinary powers of a graded radical ideal can never be Cohen-Macaulay unless it is a complete intersection. To prove the result, we introduce a new combinatorial notion called special leaf, and using this, we provide an explicit combinatorial formula of $\mathrm{depth}(R/I(Δ)^{[k]})$ for all $k\ge 1$, where $Δ$ is a Cohen-Macaulay simplicial forest. As an application, we show that the normalized depth function of a Cohen-Macaulay simplicial forest is nonincreasing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Square-free powers of Cohen-Macaulay simplicial forests Das, Kanoy Kumar Roy, Amit Saha, Kamalesh Commutative Algebra Combinatorics 13H10, 13C15 Let $I(Δ)^{[k]}$ denote the $k^{\text{th}}$ square-free power of the facet ideal of a simplicial complex $Δ$ in a polynomial ring $R$. Square-free powers are intimately related to the `Matching Theory' and `Ordinary Powers'. In this article, we show that if $Δ$ is a Cohen-Macaulay simplicial forest, then $R/I(Δ)^{[k]}$ is Cohen-Macaulay for all $k\ge 1$. This result is quite interesting since all ordinary powers of a graded radical ideal can never be Cohen-Macaulay unless it is a complete intersection. To prove the result, we introduce a new combinatorial notion called special leaf, and using this, we provide an explicit combinatorial formula of $\mathrm{depth}(R/I(Δ)^{[k]})$ for all $k\ge 1$, where $Δ$ is a Cohen-Macaulay simplicial forest. As an application, we show that the normalized depth function of a Cohen-Macaulay simplicial forest is nonincreasing. |
| title | Square-free powers of Cohen-Macaulay simplicial forests |
| topic | Commutative Algebra Combinatorics 13H10, 13C15 |
| url | https://arxiv.org/abs/2502.18396 |