Attached primes and annihilators of top local cohomology modules defined by a pair of ideals
Assume that \(R\) is a complete Noetherian local ring and \(M\) is a non-zero finitely generated \(R\)-module of dimension \(n=\dim(M)\geq 1\). It is shown that any non-empty subset \(T\) of \(\mathrm{Assh}(M)\) can be expressed as the set of attached primes of the top local cohomology modules \(H...
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Lugansk National Taras Shevchenko University
2020
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oai:ojs.admjournal.luguniv.edu.ua:article-4292020-07-08T07:13:20Z Attached primes and annihilators of top local cohomology modules defined by a pair of ideals Karimi, S. Payrovi, Sh. associated prime ideals, attached prime ideals, top local cohomology modules 13D45, 14B15 Assume that \(R\) is a complete Noetherian local ring and \(M\) is a non-zero finitely generated \(R\)-module of dimension \(n=\dim(M)\geq 1\). It is shown that any non-empty subset \(T\) of \(\mathrm{Assh}(M)\) can be expressed as the set of attached primes of the top local cohomology modules \(H_{I,J}^n(M)\) for some proper ideals \(I,J\) of \(R\). Moreover, for ideals \(I, J=\bigcap_ {\mathfrak p\in \mathrm{Att}_R(H_{I}^n(M))}\mathfrak p\) and \(J'\) of \(R\) it is proved that \(T=\mathrm{Att}_R(H_{I,J}^n(M))=\mathrm{Att}_R(H_{I,J'}^n(M))\) if and only if \(J'\subseteq J\). Let \(H_{I,J}^n(M)\neq 0\). It is shown that there exists \(Q\in \mathrm{Supp}(M)\) such that \(\dim(R/Q)=1\) and \(H_Q^n(R/{\mathfrak p})\neq 0\), for each \(\mathfrak p \in \mathrm{Att}_R(H_{I,J}^n(M))\). In addition, we prove that if \(I\) and \(J\) are two proper ideals of a Noetherian local ring \(R\), then \(\mathrm{Ann}_R(H_{I,J}^{n}(M))=\mathrm{Ann}_R(M/{T_R(I,J,M)})\), where \(T_R(I,J,M)\) is the largest submodule of \(M\) with \(\mathrm{cd}(I,J,T_R(I,J,M))<\mathrm{cd}(I,J,M)\), here \(\mathrm{cd}(I,J,M)\) is the cohomological dimension of \(M\) with respect to \(I\) and \(J\). This result is a generalization of [1, Theorem 2.3] and [2, Theorem 2.6]. Lugansk National Taras Shevchenko University 2020-07-08 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/429 10.12958/adm429 Algebra and Discrete Mathematics; Vol 29, No 2 (2020) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/429/pdf Copyright (c) 2020 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
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| datestamp_date |
2020-07-08T07:13:20Z |
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OJS |
| language |
English |
| topic |
associated prime ideals attached prime ideals top local cohomology modules 13D45 14B15 |
| spellingShingle |
associated prime ideals attached prime ideals top local cohomology modules 13D45 14B15 Karimi, S. Payrovi, Sh. Attached primes and annihilators of top local cohomology modules defined by a pair of ideals |
| topic_facet |
associated prime ideals attached prime ideals top local cohomology modules 13D45 14B15 |
| format |
Article |
| author |
Karimi, S. Payrovi, Sh. |
| author_facet |
Karimi, S. Payrovi, Sh. |
| author_sort |
Karimi, S. |
| title |
Attached primes and annihilators of top local cohomology modules defined by a pair of ideals |
| title_short |
Attached primes and annihilators of top local cohomology modules defined by a pair of ideals |
| title_full |
Attached primes and annihilators of top local cohomology modules defined by a pair of ideals |
| title_fullStr |
Attached primes and annihilators of top local cohomology modules defined by a pair of ideals |
| title_full_unstemmed |
Attached primes and annihilators of top local cohomology modules defined by a pair of ideals |
| title_sort |
attached primes and annihilators of top local cohomology modules defined by a pair of ideals |
| description |
Assume that \(R\) is a complete Noetherian local ring and \(M\) is a non-zero finitely generated \(R\)-module of dimension \(n=\dim(M)\geq 1\). It is shown that any non-empty subset \(T\) of \(\mathrm{Assh}(M)\) can be expressed as the set of attached primes of the top local cohomology modules \(H_{I,J}^n(M)\) for some proper ideals \(I,J\) of \(R\). Moreover, for ideals \(I, J=\bigcap_ {\mathfrak p\in \mathrm{Att}_R(H_{I}^n(M))}\mathfrak p\) and \(J'\) of \(R\) it is proved that \(T=\mathrm{Att}_R(H_{I,J}^n(M))=\mathrm{Att}_R(H_{I,J'}^n(M))\) if and only if \(J'\subseteq J\). Let \(H_{I,J}^n(M)\neq 0\). It is shown that there exists \(Q\in \mathrm{Supp}(M)\) such that \(\dim(R/Q)=1\) and \(H_Q^n(R/{\mathfrak p})\neq 0\), for each \(\mathfrak p \in \mathrm{Att}_R(H_{I,J}^n(M))\). In addition, we prove that if \(I\) and \(J\) are two proper ideals of a Noetherian local ring \(R\), then \(\mathrm{Ann}_R(H_{I,J}^{n}(M))=\mathrm{Ann}_R(M/{T_R(I,J,M)})\), where \(T_R(I,J,M)\) is the largest submodule of \(M\) with \(\mathrm{cd}(I,J,T_R(I,J,M))<\mathrm{cd}(I,J,M)\), here \(\mathrm{cd}(I,J,M)\) is the cohomological dimension of \(M\) with respect to \(I\) and \(J\). This result is a generalization of [1, Theorem 2.3] and [2, Theorem 2.6]. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2020 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/429 |
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2025-07-17T10:32:40Z |
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2025-07-17T10:32:40Z |
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