Finiteness of homological dimensions of Ext modules
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915479951507456 |
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| author | Kimura, Kaito |
| author_facet | Kimura, Kaito |
| contents | Let $R$ be a commutative Noetherian local ring and let $M$ and $N$ be nonzero finitely generated $R$-modules. In this paper, we investigate how the finiteness of the homological dimension of Ext modules between $M$ and $N$ affects that of $M$ and $N$. One of our main result states that if $\operatorname{Ext}^i_R(M,N)$ has finite projective dimension for any $0\le i\le \operatorname{Rfd}_R M$, where $\operatorname{Rfd}_R M$ is the (large) restricted flat dimension of $M$, then $M$ has finite projective or injective dimension if and only if $N$ does. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12843 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finiteness of homological dimensions of Ext modules Kimura, Kaito Commutative Algebra 13D05, 13D07 Let $R$ be a commutative Noetherian local ring and let $M$ and $N$ be nonzero finitely generated $R$-modules. In this paper, we investigate how the finiteness of the homological dimension of Ext modules between $M$ and $N$ affects that of $M$ and $N$. One of our main result states that if $\operatorname{Ext}^i_R(M,N)$ has finite projective dimension for any $0\le i\le \operatorname{Rfd}_R M$, where $\operatorname{Rfd}_R M$ is the (large) restricted flat dimension of $M$, then $M$ has finite projective or injective dimension if and only if $N$ does. |
| title | Finiteness of homological dimensions of Ext modules |
| topic | Commutative Algebra 13D05, 13D07 |
| url | https://arxiv.org/abs/2508.12843 |