A note on Ratliff-Rush filtration, reduction number and postulation number of $\mathfrak m$-primary ideals
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918394187481088 |
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| 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. Let rd$(I)$ be the reduction number of $I$ and n$(I)$ the postulation number. We prove that for $d=2,$ if n$(I)=ρ(I)-1,$ then rd$(I) \leq$n$(I)+2$ and if n$(I)\neq ρ(I)-1,$ then rd$(I)\geq$n$(I)+2.$ For $d \geq 3$, if $I$ is integrally closed, depth gr$(I) = d-2$ and n$(I)=-(d-3).$ Then we prove that rd$(I)\geq$n$(I)+d$. Our main result is to generalize a result of T. Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring with the good behaviour of the Ratliff-Rush filtration with respect to $I$ mod a superficial element. From this result, it follows that for a Cohen-Macaulay ring of dimension $d\geq2$, if $P_{I}(k)=H_{I}(k)$ for some $k \geq ρ(I)$, then $P_{I}(n)=H_{I}(n)$ for all $n \geq k.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_01196 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on Ratliff-Rush filtration, reduction number and postulation number of $\mathfrak m$-primary ideals 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. Let rd$(I)$ be the reduction number of $I$ and n$(I)$ the postulation number. We prove that for $d=2,$ if n$(I)=ρ(I)-1,$ then rd$(I) \leq$n$(I)+2$ and if n$(I)\neq ρ(I)-1,$ then rd$(I)\geq$n$(I)+2.$ For $d \geq 3$, if $I$ is integrally closed, depth gr$(I) = d-2$ and n$(I)=-(d-3).$ Then we prove that rd$(I)\geq$n$(I)+d$. Our main result is to generalize a result of T. Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring with the good behaviour of the Ratliff-Rush filtration with respect to $I$ mod a superficial element. From this result, it follows that for a Cohen-Macaulay ring of dimension $d\geq2$, if $P_{I}(k)=H_{I}(k)$ for some $k \geq ρ(I)$, then $P_{I}(n)=H_{I}(n)$ for all $n \geq k.$ |
| title | A note on Ratliff-Rush filtration, reduction number and postulation number of $\mathfrak m$-primary ideals |
| topic | Commutative Algebra 13H10, 13D40, 13A30 |
| url | https://arxiv.org/abs/2307.01196 |