Dao numbers and the asymptotic behaviour of fullness
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916876240551936 |
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| author | Ficarra, Antonino |
| author_facet | Ficarra, Antonino |
| contents | In the present paper, we study the Dao numbers $\mathfrak{d}_1(I),\mathfrak{d}_2(I)$ and $\mathfrak{d}_3(I)$ of an ideal $I$ of a Noetherian local ring $(R,\mathfrak{m},K)$ or a standard graded Noetherian $K$-algebra. They are defined as the smallest $\ell\ge0$ such that $I\mathfrak{m}^k$ is $\mathfrak{m}$-full, full, weakly $\mathfrak{m}$-full, respectively, for all $k\ge\ell$. We provide general bounds for the Dao numbers in terms of the Castelnuovo-Mumford regularity of certain modules over the Rees algebra $\mathcal{R}(\mathfrak{m})$. If $R$ is a Koszul algebra, we prove that the Dao numbers are less or equal to $\text{reg}_{\text{gr}_\mathfrak{m}(R)}\text{gr}_\mathfrak{m}(I)$, where $\text{gr}_\mathfrak{m}(I)$ is the associated graded module of $I$. Finally, for monomial ideals, we combinatorially bound the Dao numbers in terms of asymptotic linear quotients and bounding multidegrees. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_05555 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dao numbers and the asymptotic behaviour of fullness Ficarra, Antonino Commutative Algebra Combinatorics In the present paper, we study the Dao numbers $\mathfrak{d}_1(I),\mathfrak{d}_2(I)$ and $\mathfrak{d}_3(I)$ of an ideal $I$ of a Noetherian local ring $(R,\mathfrak{m},K)$ or a standard graded Noetherian $K$-algebra. They are defined as the smallest $\ell\ge0$ such that $I\mathfrak{m}^k$ is $\mathfrak{m}$-full, full, weakly $\mathfrak{m}$-full, respectively, for all $k\ge\ell$. We provide general bounds for the Dao numbers in terms of the Castelnuovo-Mumford regularity of certain modules over the Rees algebra $\mathcal{R}(\mathfrak{m})$. If $R$ is a Koszul algebra, we prove that the Dao numbers are less or equal to $\text{reg}_{\text{gr}_\mathfrak{m}(R)}\text{gr}_\mathfrak{m}(I)$, where $\text{gr}_\mathfrak{m}(I)$ is the associated graded module of $I$. Finally, for monomial ideals, we combinatorially bound the Dao numbers in terms of asymptotic linear quotients and bounding multidegrees. |
| title | Dao numbers and the asymptotic behaviour of fullness |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2402.05555 |