Algebraic properties of tensor product of modules over a field
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916705461075968 |
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| author | Rahimi, Ahad |
| author_facet | Rahimi, Ahad |
| contents | Let $A$ and $B$ be commutative Noetherian algebras over an arbitrary field $\Bbbk$ such that $A \otimes_\Bbbk B$ is Noetherian. We consider ideals $I$ and $J$ of $A$ and $B$, respectively, as well as nonzero finitely generated modules $L$ and $N$ over $A$ and $B$, respectively. In this paper, we investigate certain algebraic properties of the $A \otimes_\Bbbk B$-module $L\otimes_{\Bbbk} N$, which are often inherited from the properties of the $A$-module $L$ and the $B$-module $N$. Specifically, we provide characterizations for the Cohen-Macaulayness, generalized Cohen-Macaulayness, and sequentially Cohen-Macaulayness of $L\otimes_{\Bbbk} N$ with respect to the ideal $I \otimes_\Bbbk B + A \otimes_\Bbbk J$, in terms of the corresponding properties for $L$ and $N$ with respect to $I$ and $J$, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17217 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic properties of tensor product of modules over a field Rahimi, Ahad Commutative Algebra 13C14, 13C15, 18G40, 13D45 Let $A$ and $B$ be commutative Noetherian algebras over an arbitrary field $\Bbbk$ such that $A \otimes_\Bbbk B$ is Noetherian. We consider ideals $I$ and $J$ of $A$ and $B$, respectively, as well as nonzero finitely generated modules $L$ and $N$ over $A$ and $B$, respectively. In this paper, we investigate certain algebraic properties of the $A \otimes_\Bbbk B$-module $L\otimes_{\Bbbk} N$, which are often inherited from the properties of the $A$-module $L$ and the $B$-module $N$. Specifically, we provide characterizations for the Cohen-Macaulayness, generalized Cohen-Macaulayness, and sequentially Cohen-Macaulayness of $L\otimes_{\Bbbk} N$ with respect to the ideal $I \otimes_\Bbbk B + A \otimes_\Bbbk J$, in terms of the corresponding properties for $L$ and $N$ with respect to $I$ and $J$, respectively. |
| title | Algebraic properties of tensor product of modules over a field |
| topic | Commutative Algebra 13C14, 13C15, 18G40, 13D45 |
| url | https://arxiv.org/abs/2504.17217 |