Strong persistence index and fluctuations in colon powers of monomial ideals
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911706450493440 |
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| author | Nasernejad, Mehrdad Toledo, Jonathan |
| author_facet | Nasernejad, Mehrdad Toledo, Jonathan |
| contents | Let $I$ be an ideal in a commutative Noetherian ring $R$. We say that a positive integer $\ell_0$ is the strong persistence index of $I$ if $\ell_0$ is the smallest integer such that $(I^{\ell+1} :_R I) = I^{\ell}$ for all $\ell \geq \ell_0$. The first aim of this paper is to study this notion for monomial ideals.
We also introduce the notion of fluctuation in colon powers if there exist positive integers $a < b < c$ such that at least one of the following cases occurs:
(i) $(I^{a} : I) = I^{a-1}$, $(I^{b} : I) \neq I^{b-1}$, but $(I^{c} : I) = I^{c-1}$.
(ii) $(I^{a} : I) \neq I^{a-1}$, $(I^{b} : I) = I^{b-1}$, but $(I^{c} : I) \neq I^{c-1}$.
The second purpose of this work is to study this phenomenon for monomial ideals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11475 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strong persistence index and fluctuations in colon powers of monomial ideals Nasernejad, Mehrdad Toledo, Jonathan Commutative Algebra Let $I$ be an ideal in a commutative Noetherian ring $R$. We say that a positive integer $\ell_0$ is the strong persistence index of $I$ if $\ell_0$ is the smallest integer such that $(I^{\ell+1} :_R I) = I^{\ell}$ for all $\ell \geq \ell_0$. The first aim of this paper is to study this notion for monomial ideals. We also introduce the notion of fluctuation in colon powers if there exist positive integers $a < b < c$ such that at least one of the following cases occurs: (i) $(I^{a} : I) = I^{a-1}$, $(I^{b} : I) \neq I^{b-1}$, but $(I^{c} : I) = I^{c-1}$. (ii) $(I^{a} : I) \neq I^{a-1}$, $(I^{b} : I) = I^{b-1}$, but $(I^{c} : I) \neq I^{c-1}$. The second purpose of this work is to study this phenomenon for monomial ideals. |
| title | Strong persistence index and fluctuations in colon powers of monomial ideals |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2604.11475 |