Some characterizations of Gorenstein Rees Algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913368423530496 |
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| author | Iai, Shin-ichiro |
| author_facet | Iai, Shin-ichiro |
| contents | The aim of this paper is to elucidate the relationship between the Gorenstein Rees algebra $\R(I):=\bigoplus_{i\ge 0}I^i$ of an ideal $I$ in a complete Noetherian local ring $A$ and the graded canonical module of the extended Rees algebra $\R'(I):=\bigoplus_{i\in\Z}I^i$. It is known that the Gorensteinness of $\R(I)$ is closely related to the property of the graded canonical module of the associated graded ring $\G(I):=\bigoplus_{i\ge 0}I^i/I^{i+1}$. However, there appears to be a shortage of satisfactory references analyzing the relationship between $\R(I)$ and $\R'(I)$ unless the ring $\G(I)$ is Cohen-Macaulay. This paper provides a characterization of the Gorenstein property of $\R(I)$ using the graded canonical module of $\R'(I)$ without assuming that the base ring $A$ is Cohen-Macaulay. Applying our criterion, we demonstrate that a certain Kawasaki's arithmetic Cohen-Macaulayfication becomes a Gorenstein ring when $A$ is a quasi-Gorenstein local ring with finite local cohomology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18963 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some characterizations of Gorenstein Rees Algebras Iai, Shin-ichiro Commutative Algebra The aim of this paper is to elucidate the relationship between the Gorenstein Rees algebra $\R(I):=\bigoplus_{i\ge 0}I^i$ of an ideal $I$ in a complete Noetherian local ring $A$ and the graded canonical module of the extended Rees algebra $\R'(I):=\bigoplus_{i\in\Z}I^i$. It is known that the Gorensteinness of $\R(I)$ is closely related to the property of the graded canonical module of the associated graded ring $\G(I):=\bigoplus_{i\ge 0}I^i/I^{i+1}$. However, there appears to be a shortage of satisfactory references analyzing the relationship between $\R(I)$ and $\R'(I)$ unless the ring $\G(I)$ is Cohen-Macaulay. This paper provides a characterization of the Gorenstein property of $\R(I)$ using the graded canonical module of $\R'(I)$ without assuming that the base ring $A$ is Cohen-Macaulay. Applying our criterion, we demonstrate that a certain Kawasaki's arithmetic Cohen-Macaulayfication becomes a Gorenstein ring when $A$ is a quasi-Gorenstein local ring with finite local cohomology. |
| title | Some characterizations of Gorenstein Rees Algebras |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2405.18963 |