Bass numbers and endomorphism rings of Gorenstein injective modules
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914708066402304 |
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| author | Sazeedeh, Reza |
| author_facet | Sazeedeh, Reza |
| contents | Let $R$ be a commutative noetherian ring admitting a dualizing complex and let $\mathfrak p$ be a prime ideal of $R$. In this paper we investigate when $G(R/\frak p)$ is an $R_{\frak p}$-module. We give some necessary and sufficient conditions under which $G(R/\frak p)$ is an $R_{\frak p}$-module. We also study the Bass numbers of $G(R/\frak p)$ and we show that if ${\rm Gid}_RR/\frak p$ is finite, then $μ^i(\frak q,G(R/\frak p))$ is finite for all $i\geq 0$ and all $\frak q\in{\rm Spec} R$. If ${\rm Gpd}_RR/\frak p$ is finite, then $μ^i(\frak p,G(R/\frak p))$ is finite for all $i\geq 0$. We define a subring $S(\frak p)_{\frak p}$ of ${\rm End}_{R_{\frak p}}(G(R_{\frak p}/\frak pR_{\frak p}))$ and we show that it is noetherian and contains a subring which is a quotient of $\widehat{R_{\frak p}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05207 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bass numbers and endomorphism rings of Gorenstein injective modules Sazeedeh, Reza Commutative Algebra 13D02, 13D45, 18G25 Let $R$ be a commutative noetherian ring admitting a dualizing complex and let $\mathfrak p$ be a prime ideal of $R$. In this paper we investigate when $G(R/\frak p)$ is an $R_{\frak p}$-module. We give some necessary and sufficient conditions under which $G(R/\frak p)$ is an $R_{\frak p}$-module. We also study the Bass numbers of $G(R/\frak p)$ and we show that if ${\rm Gid}_RR/\frak p$ is finite, then $μ^i(\frak q,G(R/\frak p))$ is finite for all $i\geq 0$ and all $\frak q\in{\rm Spec} R$. If ${\rm Gpd}_RR/\frak p$ is finite, then $μ^i(\frak p,G(R/\frak p))$ is finite for all $i\geq 0$. We define a subring $S(\frak p)_{\frak p}$ of ${\rm End}_{R_{\frak p}}(G(R_{\frak p}/\frak pR_{\frak p}))$ and we show that it is noetherian and contains a subring which is a quotient of $\widehat{R_{\frak p}}$. |
| title | Bass numbers and endomorphism rings of Gorenstein injective modules |
| topic | Commutative Algebra 13D02, 13D45, 18G25 |
| url | https://arxiv.org/abs/2403.05207 |