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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.15368 |
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| _version_ | 1866916753554014208 |
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| author | Gimenez, Philippe Planas-Vilanova, Francesc |
| author_facet | Gimenez, Philippe Planas-Vilanova, Francesc |
| contents | Let $A$ be a noetherian ring, $I$ an ideal of $A$ and $N\subset M$ finitely generated $A$-modules. The relation type of $I$ with respect to $M$, denoted by ${\bf rt}\,(I;M)$, is the maximal degree in a minimal generating set of relations of the Rees module ${\bf R}(I;M)=\oplus_{n\geq 0}I^nM$. It is a well-known invariant that gives a first measure of the complexity of ${\bf R}(I;M)$. To help to measure this complexity, we introduce the sifted type of ${\bf R}(I;M)$, denoted by ${\bf st}\,(I;M)$, a new invariant which counts the non-zero degrees appearing in a minimal generating set of relations of ${\bf R}(I;M)$. Just as the relation type ${\bf rt}\,(I;M/N)$ is closely related to the strong Artin-Rees number ${\bf s}\,(N,M;I)$, it turns out that the sifted type ${\bf st}\,(I;M/N)$ is closely related to the medium Artin-Rees number ${\bf m}\,(N,M;I)$, a new invariant which lies in between the weak and the strong Artin-Rees numbers of $(N,M;I)$. We illustrate the meaning, interest and mutual connection of ${\bf m}\,(N,M;I)$ and ${\bf st}\,(I;M)$ with some examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sifted degrees of the equations of the Rees module and their connection with the Artin-Rees numbers Gimenez, Philippe Planas-Vilanova, Francesc Commutative Algebra 13A30, 13D02 Let $A$ be a noetherian ring, $I$ an ideal of $A$ and $N\subset M$ finitely generated $A$-modules. The relation type of $I$ with respect to $M$, denoted by ${\bf rt}\,(I;M)$, is the maximal degree in a minimal generating set of relations of the Rees module ${\bf R}(I;M)=\oplus_{n\geq 0}I^nM$. It is a well-known invariant that gives a first measure of the complexity of ${\bf R}(I;M)$. To help to measure this complexity, we introduce the sifted type of ${\bf R}(I;M)$, denoted by ${\bf st}\,(I;M)$, a new invariant which counts the non-zero degrees appearing in a minimal generating set of relations of ${\bf R}(I;M)$. Just as the relation type ${\bf rt}\,(I;M/N)$ is closely related to the strong Artin-Rees number ${\bf s}\,(N,M;I)$, it turns out that the sifted type ${\bf st}\,(I;M/N)$ is closely related to the medium Artin-Rees number ${\bf m}\,(N,M;I)$, a new invariant which lies in between the weak and the strong Artin-Rees numbers of $(N,M;I)$. We illustrate the meaning, interest and mutual connection of ${\bf m}\,(N,M;I)$ and ${\bf st}\,(I;M)$ with some examples. |
| title | Sifted degrees of the equations of the Rees module and their connection with the Artin-Rees numbers |
| topic | Commutative Algebra 13A30, 13D02 |
| url | https://arxiv.org/abs/2412.15368 |