Depth and Stanley depth of powers of the path ideal of a cycle graph
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909091612327936 |
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| author | Balanescu, Silviu Cimpoeas, Mircea |
| author_facet | Balanescu, Silviu Cimpoeas, Mircea |
| contents | Let $J_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\; x_{n-m+1}\cdots x_n,\; x_{n-m+2}\cdots x_nx_1, \ldots, x_nx_1\cdots x_{m-1})$ be the $m$-path ideal of the cycle graph of length $n$, in the ring $S=K[x_1,\ldots,x_n]$.
Let $d=\gcd(n,m)$. We prove that $\operatorname{depth}(S/J_{n,m}^t)\leq d-1$ for all $t\geq n-1$. We show that $\operatorname{sdepth}(S/J_{n,n-1}^t)=\operatorname{depth}(S/J_{n,n-1}^t)=\max\{n-t-1,0\}$ for all $t\geq 1$. Also, we give some bounds for $\operatorname{depth}(S/J_{n,m}^t)$ and $\operatorname{sdepth}(S/J_{n,m}^t)$, where $t\geq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_15032 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Depth and Stanley depth of powers of the path ideal of a cycle graph Balanescu, Silviu Cimpoeas, Mircea Commutative Algebra 13C15, 13P10, 13F20 Let $J_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\; x_{n-m+1}\cdots x_n,\; x_{n-m+2}\cdots x_nx_1, \ldots, x_nx_1\cdots x_{m-1})$ be the $m$-path ideal of the cycle graph of length $n$, in the ring $S=K[x_1,\ldots,x_n]$. Let $d=\gcd(n,m)$. We prove that $\operatorname{depth}(S/J_{n,m}^t)\leq d-1$ for all $t\geq n-1$. We show that $\operatorname{sdepth}(S/J_{n,n-1}^t)=\operatorname{depth}(S/J_{n,n-1}^t)=\max\{n-t-1,0\}$ for all $t\geq 1$. Also, we give some bounds for $\operatorname{depth}(S/J_{n,m}^t)$ and $\operatorname{sdepth}(S/J_{n,m}^t)$, where $t\geq 1$. |
| title | Depth and Stanley depth of powers of the path ideal of a cycle graph |
| topic | Commutative Algebra 13C15, 13P10, 13F20 |
| url | https://arxiv.org/abs/2303.15032 |