Local cohomology modules of a regular affine domain
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915606147629056 |
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| author | Islam, Sayed Sadiqul |
| author_facet | Islam, Sayed Sadiqul |
| contents | For a Noetherian commutative ring $R$, let $H^i_I(R)$ be the $ i$-th local cohomology module of $R$ with respect to $I$. In \cite{Hel-08}, Hellus posed the question of identifying rings $R$ such that $\operatorname{injdim}_R H^i_I(R)=\operatorname{dim}_R(\operatorname{Supp}_R H^i_I(R))$. In this paper, we show that a regular affine domain over a field of characteristic $0$ satisfies this condition. In fact, we prove that $\operatorname{injdim}_R H^i_I(R)\geq \operatorname{dim}_R(\operatorname{Supp}_R H^i_I(R))-1$ when $R$ is a differentiably admissible $K$-algebra. Indeed, we establish both of these conclusions for a substantially broad class of functors known as Lyubeznik functors. We also prove that if $R$ is a polynomial ring over a differentiably admissible $K$-algebra, then $\operatorname{Ass}_R H^i_I(R)$ is finite for all $i\geq 0$ and for every ideal $I$ of $R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05871 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local cohomology modules of a regular affine domain Islam, Sayed Sadiqul Commutative Algebra Primary 13D45, Secondary 13N10, 13H10 For a Noetherian commutative ring $R$, let $H^i_I(R)$ be the $ i$-th local cohomology module of $R$ with respect to $I$. In \cite{Hel-08}, Hellus posed the question of identifying rings $R$ such that $\operatorname{injdim}_R H^i_I(R)=\operatorname{dim}_R(\operatorname{Supp}_R H^i_I(R))$. In this paper, we show that a regular affine domain over a field of characteristic $0$ satisfies this condition. In fact, we prove that $\operatorname{injdim}_R H^i_I(R)\geq \operatorname{dim}_R(\operatorname{Supp}_R H^i_I(R))-1$ when $R$ is a differentiably admissible $K$-algebra. Indeed, we establish both of these conclusions for a substantially broad class of functors known as Lyubeznik functors. We also prove that if $R$ is a polynomial ring over a differentiably admissible $K$-algebra, then $\operatorname{Ass}_R H^i_I(R)$ is finite for all $i\geq 0$ and for every ideal $I$ of $R$. |
| title | Local cohomology modules of a regular affine domain |
| topic | Commutative Algebra Primary 13D45, Secondary 13N10, 13H10 |
| url | https://arxiv.org/abs/2511.05871 |