Bounds on the Ratliff-Rush Index and the Castelnuovo-Mumford Regularity
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908891677196288 |
|---|---|
| author | Mandal, Mousumi Priya, Shruti |
| author_facet | Mandal, Mousumi Priya, Shruti |
| contents | Let $(R, \mathfrak m)$ be a Cohen-Macaulay local ring of dimension $d \geq 2,$ and $I$ an $\mathfrak m$-primary ideal of $R.$ Denote $r_{J}(I)$ as the reduction number of $I$ with respect to a minimal reduction $J$ of $I,$ and $ρ(I)$ as the Ratliff-Rush index of $I$. We establish upper bounds on $ρ(I)$ in terms of Hilbert coefficients $e_{i}(I)$ for $0 \leq i \leq d+1,$ and $r_{J}(I).$ Suppose $\widetilde{I^{r_{J}(I)}} \neq I^{r_{J}(I)}.$ We prove that $ρ(I) \leq r_{J}(I)-1+(-1)^{d+1}(e_{d+1}(I)-\widetilde{e}_{d+1}(I)).$ When $d=2,$ we prove that $ρ(I) \leq r_{J}(I) -1 +(e_{2}(I)-1)e_{2}(I)-e_{3}(I).$ This established bound on $ρ(I)$ consequently leads to a bound on the Castelnuovo-Mumford regularity of the associated graded ring of $I.$ We also determine bound on $ρ(I)$ in two-dimensional Buchsbaum rings with positive depth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01684 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on the Ratliff-Rush Index and the Castelnuovo-Mumford Regularity Mandal, Mousumi Priya, Shruti Commutative Algebra 13H10, 13D40, 13A30 Let $(R, \mathfrak m)$ be a Cohen-Macaulay local ring of dimension $d \geq 2,$ and $I$ an $\mathfrak m$-primary ideal of $R.$ Denote $r_{J}(I)$ as the reduction number of $I$ with respect to a minimal reduction $J$ of $I,$ and $ρ(I)$ as the Ratliff-Rush index of $I$. We establish upper bounds on $ρ(I)$ in terms of Hilbert coefficients $e_{i}(I)$ for $0 \leq i \leq d+1,$ and $r_{J}(I).$ Suppose $\widetilde{I^{r_{J}(I)}} \neq I^{r_{J}(I)}.$ We prove that $ρ(I) \leq r_{J}(I)-1+(-1)^{d+1}(e_{d+1}(I)-\widetilde{e}_{d+1}(I)).$ When $d=2,$ we prove that $ρ(I) \leq r_{J}(I) -1 +(e_{2}(I)-1)e_{2}(I)-e_{3}(I).$ This established bound on $ρ(I)$ consequently leads to a bound on the Castelnuovo-Mumford regularity of the associated graded ring of $I.$ We also determine bound on $ρ(I)$ in two-dimensional Buchsbaum rings with positive depth. |
| title | Bounds on the Ratliff-Rush Index and the Castelnuovo-Mumford Regularity |
| topic | Commutative Algebra 13H10, 13D40, 13A30 |
| url | https://arxiv.org/abs/2404.01684 |