Bounded powers of edge ideals: Gorenstein toric rings
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912405036990464 |
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| author | Hibi, Takayuki Fakhari, Seyed Amin Seyed |
| author_facet | Hibi, Takayuki Fakhari, Seyed Amin Seyed |
| contents | Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I \subset S$ a monomial ideal. Given a vector $\mathfrak{c}\in\mathbb{N}^n$, the ideal $I_{\mathfrak{c}}$ is the ideal generated by those monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Let $δ_{\mathfrak{c}}(I)$ be the largest integer $q$ for which $(I^q)_{\mathfrak{c}}\neq 0$. For a finite graph $G$, its edge ideal is denoted by $I(G)$. Let $\mathcal{B}(\mathfrak{c},G)$ be the toric ring which is generated by the monomials belonging to the minimal system of monomial generators of $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$. In a previous work, the authors proved that $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. It follows that $\mathcal{B}(\mathfrak{c},G)$ is a normal Cohen--Macaulay domain. In this paper, we study the Gorenstein property of $\mathcal{B}(\mathfrak{c},G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21760 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded powers of edge ideals: Gorenstein toric rings Hibi, Takayuki Fakhari, Seyed Amin Seyed Commutative Algebra Combinatorics Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I \subset S$ a monomial ideal. Given a vector $\mathfrak{c}\in\mathbb{N}^n$, the ideal $I_{\mathfrak{c}}$ is the ideal generated by those monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Let $δ_{\mathfrak{c}}(I)$ be the largest integer $q$ for which $(I^q)_{\mathfrak{c}}\neq 0$. For a finite graph $G$, its edge ideal is denoted by $I(G)$. Let $\mathcal{B}(\mathfrak{c},G)$ be the toric ring which is generated by the monomials belonging to the minimal system of monomial generators of $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$. In a previous work, the authors proved that $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. It follows that $\mathcal{B}(\mathfrak{c},G)$ is a normal Cohen--Macaulay domain. In this paper, we study the Gorenstein property of $\mathcal{B}(\mathfrak{c},G)$. |
| title | Bounded powers of edge ideals: Gorenstein toric rings |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2504.21760 |