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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2401.02751 |
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| _version_ | 1866917559544053760 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $A$ be a Noetherian ring and let $\mathcal{R} = \bigoplus_{n \geq 0}\mathcal{R}_n$ be a standard graded ring with $\mathcal{R}_0 = A$. We define a category $\mathfrak{A}(\mathcal{R})$ of graded $\mathcal{R}$-modules (not necessarily finitely generated) with the following properties: if $X = \bigoplus_{n \in \mathbb{Z}} X_n \in \mathfrak{A}(\mathcal{R}) $ then
(1) $X_i$ is finitely generated $A$-module for all $i \in \mathbb{Z}$ and $X_i = 0$ for $i \ll 0$.
(2) There exists $n_0$ such that $\text{Ass}_A X_n = \text{Ass}_A X_{n_0}$ for all $n \geq n_0$.
(3) If $X_n$ has finite length as an $A$-module for all $n$ then there exists $P_X(z) \in \mathbb{Q}[z]$ such that $P_X(n) = \ell_A(X_n)$ for all $n \gg 0$.
(4) If $F$ is a coherent functor on the category of finitely generated $A$-modules then $F(X) = \bigoplus_{n \in \mathbb{Z}} F(X_n) \in \mathfrak{A}(\mathcal{R})$.
(5) For an ideal $J$ in $A$, there exists $c_J^X$ such that $\text{grade}(J, X_n) = \text{grade}(J, X_{c_J^X})$ for all $n \geq c_J^X$.
We give a unified proof of several results in theory of associate primes and related areas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02751 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A convenient category to study asymptotic primes and related questions Puthenpurakal, Tony J. Commutative Algebra Primary 13A30, Secondary 13E05, 18A22 Let $A$ be a Noetherian ring and let $\mathcal{R} = \bigoplus_{n \geq 0}\mathcal{R}_n$ be a standard graded ring with $\mathcal{R}_0 = A$. We define a category $\mathfrak{A}(\mathcal{R})$ of graded $\mathcal{R}$-modules (not necessarily finitely generated) with the following properties: if $X = \bigoplus_{n \in \mathbb{Z}} X_n \in \mathfrak{A}(\mathcal{R}) $ then (1) $X_i$ is finitely generated $A$-module for all $i \in \mathbb{Z}$ and $X_i = 0$ for $i \ll 0$. (2) There exists $n_0$ such that $\text{Ass}_A X_n = \text{Ass}_A X_{n_0}$ for all $n \geq n_0$. (3) If $X_n$ has finite length as an $A$-module for all $n$ then there exists $P_X(z) \in \mathbb{Q}[z]$ such that $P_X(n) = \ell_A(X_n)$ for all $n \gg 0$. (4) If $F$ is a coherent functor on the category of finitely generated $A$-modules then $F(X) = \bigoplus_{n \in \mathbb{Z}} F(X_n) \in \mathfrak{A}(\mathcal{R})$. (5) For an ideal $J$ in $A$, there exists $c_J^X$ such that $\text{grade}(J, X_n) = \text{grade}(J, X_{c_J^X})$ for all $n \geq c_J^X$. We give a unified proof of several results in theory of associate primes and related areas. |
| title | A convenient category to study asymptotic primes and related questions |
| topic | Commutative Algebra Primary 13A30, Secondary 13E05, 18A22 |
| url | https://arxiv.org/abs/2401.02751 |