Nearly Gorenstein normal graded rings
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918322073763840 |
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| author | Okuma, Tomohiro Watanabe, Kei-ichi Yoshida, Ken-ichi |
| author_facet | Okuma, Tomohiro Watanabe, Kei-ichi Yoshida, Ken-ichi |
| contents | We investigate nearly Gorenstein property for a normal graded ring $R = \bigoplus_{n\ge 0}R_n$ finitely generated over a field. For that purpose, we investigate ${K_R}^{-1}$, the inverse of $K_R$ (the canonical module of $R$) and introduce a new invariant $b(R)$ of $R$. We investigate nearly Gorenstein property of $R$ using $a(R)$ and $b(R)$ and $m(R)$, the initial degree of $R$. If $b(R)<0$, (and if $R$ is $\mathbb Q$-Gorenstein), then we believe that $R$ is log-terminal -- this is proved if $\dim R=2$ or $R$ is F-pure (or $F$-pure type). Then we determine the condition for a $2$-dimensional cone singularity over a smooth curve of genus $g\le 3$ to be nearly Gorenstein. We observe that ``almost Gorenstein" property and nearly Gorenstein property are drastically different for such rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04222 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nearly Gorenstein normal graded rings Okuma, Tomohiro Watanabe, Kei-ichi Yoshida, Ken-ichi Commutative Algebra Primary: 13A02, Secondary: 14J17, 14B05, 13H10, 13G05 13G05 We investigate nearly Gorenstein property for a normal graded ring $R = \bigoplus_{n\ge 0}R_n$ finitely generated over a field. For that purpose, we investigate ${K_R}^{-1}$, the inverse of $K_R$ (the canonical module of $R$) and introduce a new invariant $b(R)$ of $R$. We investigate nearly Gorenstein property of $R$ using $a(R)$ and $b(R)$ and $m(R)$, the initial degree of $R$. If $b(R)<0$, (and if $R$ is $\mathbb Q$-Gorenstein), then we believe that $R$ is log-terminal -- this is proved if $\dim R=2$ or $R$ is F-pure (or $F$-pure type). Then we determine the condition for a $2$-dimensional cone singularity over a smooth curve of genus $g\le 3$ to be nearly Gorenstein. We observe that ``almost Gorenstein" property and nearly Gorenstein property are drastically different for such rings. |
| title | Nearly Gorenstein normal graded rings |
| topic | Commutative Algebra Primary: 13A02, Secondary: 14J17, 14B05, 13H10, 13G05 13G05 |
| url | https://arxiv.org/abs/2602.04222 |