Powers of facet ideals of simplicial trees
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866911372792561664 |
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| author | Kumar, Ajay Kumar, Arvind |
| author_facet | Kumar, Ajay Kumar, Arvind |
| contents | In this article, we study the linearity of the minimal free resolution of powers of facets ideals of simplicial trees. We give a complete characterization of simplicial trees for which (some) power of its facet ideal has a linear resolution. We calculate the regularity of the $t$-path ideal of a perfect rooted tree. We also obtain an upper bound for the regularity of the $t$-path ideal of a rooted tree. We give a procedure to calculate the regularity of powers of facet ideals of simplicial trees. As a consequence of this result, we study the regularity of powers of $t$-path ideals of rooted trees. We pose a regularity upper bound conjecture for facet ideals of simplicial trees, which is as follows: if $Δ$ is a $d$-dimensional simplicial tree, then $\reg(I(Δ)^s) \leq (d+1)(s-1)+\reg(I(Δ))$ for all $s \geq 1$. We prove this conjecture for some special classes of simplicial trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_01994 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Powers of facet ideals of simplicial trees Kumar, Ajay Kumar, Arvind Commutative Algebra Combinatorics In this article, we study the linearity of the minimal free resolution of powers of facets ideals of simplicial trees. We give a complete characterization of simplicial trees for which (some) power of its facet ideal has a linear resolution. We calculate the regularity of the $t$-path ideal of a perfect rooted tree. We also obtain an upper bound for the regularity of the $t$-path ideal of a rooted tree. We give a procedure to calculate the regularity of powers of facet ideals of simplicial trees. As a consequence of this result, we study the regularity of powers of $t$-path ideals of rooted trees. We pose a regularity upper bound conjecture for facet ideals of simplicial trees, which is as follows: if $Δ$ is a $d$-dimensional simplicial tree, then $\reg(I(Δ)^s) \leq (d+1)(s-1)+\reg(I(Δ))$ for all $s \geq 1$. We prove this conjecture for some special classes of simplicial trees. |
| title | Powers of facet ideals of simplicial trees |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2306.01994 |