The Graded Betti Numbers of the Skeletons of Simplicial Complexes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910056816050176 |
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| author | Namiq, Mohammed Rafiq |
| author_facet | Namiq, Mohammed Rafiq |
| contents | In this paper, we study a class $\mathcal{C}$ of squarefree monomial ideals $I\subseteq R=\mathbb{K}[x_1,\dots,x_n]$ over a field $\mathbb{K}$, defined by the condition that $\dim R/I$ equals the maximum degree of the minimal generators of $I$ minus one. We show that the Stanley-Reisner ideal of every $i$-skeleton of a simplicial complex $Δ$ belongs to $\mathcal{C}$ for all $-1\le i<\dimΔ$. To investigate their homological properties, we introduce the notion of a degree resolution and prove that every ideal in $\mathcal{C}$ possesses this property. Moreover, we show that every squarefree monomial ideal admits a truncation whose regularity coincides with that of the original ideal, thereby reducing the study of degree resolutions to that of linear resolutions. Finally, we provide an explicit formula describing the relationship between the graded Betti numbers of a simplicial complex and those of its skeletons. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_09969 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Graded Betti Numbers of the Skeletons of Simplicial Complexes Namiq, Mohammed Rafiq Commutative Algebra Combinatorics Primary 13D02, 13F55, Secondary 13A02, 55U05 In this paper, we study a class $\mathcal{C}$ of squarefree monomial ideals $I\subseteq R=\mathbb{K}[x_1,\dots,x_n]$ over a field $\mathbb{K}$, defined by the condition that $\dim R/I$ equals the maximum degree of the minimal generators of $I$ minus one. We show that the Stanley-Reisner ideal of every $i$-skeleton of a simplicial complex $Δ$ belongs to $\mathcal{C}$ for all $-1\le i<\dimΔ$. To investigate their homological properties, we introduce the notion of a degree resolution and prove that every ideal in $\mathcal{C}$ possesses this property. Moreover, we show that every squarefree monomial ideal admits a truncation whose regularity coincides with that of the original ideal, thereby reducing the study of degree resolutions to that of linear resolutions. Finally, we provide an explicit formula describing the relationship between the graded Betti numbers of a simplicial complex and those of its skeletons. |
| title | The Graded Betti Numbers of the Skeletons of Simplicial Complexes |
| topic | Commutative Algebra Combinatorics Primary 13D02, 13F55, Secondary 13A02, 55U05 |
| url | https://arxiv.org/abs/2510.09969 |