The homological shift algebra of a monomial ideal
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917988303634432 |
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| author | Ficarra, Antonino Qureshi, Ayesha Asloob |
| author_facet | Ficarra, Antonino Qureshi, Ayesha Asloob |
| contents | Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$, and let $I\subset S$ be a monomial ideal. In this paper, we introduce the $i$th \textit{homological shift algebras} $\text{HS}_i(\mathcal{R}(I))=\bigoplus_{k\ge1}\text{HS}_i(I^k)$ of $I$. If $I$ has linear powers, these algebras have the structure of a finitely generated bigraded module over the Rees algebra $\mathcal{R}(I)$ of $I$. Hence, many invariants of $\text{HS}_i(I^k)$, such as depth, associated primes, regularity, and the $\text{v}$-number, exhibit well behaved asymptotic behavior. We determine several families of monomial ideals $I$ for which $\text{HS}_i(I^k)$ has linear resolution for all $k\gg0$. Finally, we show that $\text{HS}_i(I^k)$ is Golod for all monomial ideals $I\subset S$ with linear powers and all $k\gg0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21031 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The homological shift algebra of a monomial ideal Ficarra, Antonino Qureshi, Ayesha Asloob Commutative Algebra Combinatorics Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$, and let $I\subset S$ be a monomial ideal. In this paper, we introduce the $i$th \textit{homological shift algebras} $\text{HS}_i(\mathcal{R}(I))=\bigoplus_{k\ge1}\text{HS}_i(I^k)$ of $I$. If $I$ has linear powers, these algebras have the structure of a finitely generated bigraded module over the Rees algebra $\mathcal{R}(I)$ of $I$. Hence, many invariants of $\text{HS}_i(I^k)$, such as depth, associated primes, regularity, and the $\text{v}$-number, exhibit well behaved asymptotic behavior. We determine several families of monomial ideals $I$ for which $\text{HS}_i(I^k)$ has linear resolution for all $k\gg0$. Finally, we show that $\text{HS}_i(I^k)$ is Golod for all monomial ideals $I\subset S$ with linear powers and all $k\gg0$. |
| title | The homological shift algebra of a monomial ideal |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2412.21031 |