Geodesic
On the minimaxness and coatomicness of local cohomology modules
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 1, pp. 177-190
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Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $\mathcal {C}$-minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$-module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that $(H^i_I(M))_{\frak m}$ is a minimax $R_{\frak m}$-module for all $\frak m \in {\rm Max} (R)$ and for all $i n$, then the set ${\rm Ass}_R(H^n_I(M))$ is finite. Also, if $H^i_I(M)$ is minimax for all $i \geq n \geq 1$, then $H^i_I(M)$ is Artinian for $i \geq n$. It is shown that if $M$ is a $\mathcal {C}$-minimax module over a local ring such that $H^i_I(M)$ are $\mathcal {C}$-minimax modules for all $i n$ (or $i\geq n$), where $n\geq 1$, then they must be minimax. Consequently, a vanishing theorem is proved for local cohomology modules.
Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $\mathcal {C}$-minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$-module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that $(H^i_I(M))_{\frak m}$ is a minimax $R_{\frak m}$-module for all $\frak m \in {\rm Max} (R)$ and for all $i n$, then the set ${\rm Ass}_R(H^n_I(M))$ is finite. Also, if $H^i_I(M)$ is minimax for all $i \geq n \geq 1$, then $H^i_I(M)$ is Artinian for $i \geq n$. It is shown that if $M$ is a $\mathcal {C}$-minimax module over a local ring such that $H^i_I(M)$ are $\mathcal {C}$-minimax modules for all $i n$ (or $i\geq n$), where $n\geq 1$, then they must be minimax. Consequently, a vanishing theorem is proved for local cohomology modules.
DOI : 10.21136/CMJ.2021.0383-20
Classification : 13C05, 13D45, 13E05
Keywords: local cohomology module; minimax module; coatomic module; Artinian module; local-global principle 
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Hatamkhani, Marzieh; Roshan-Shekalgourabi, Hajar. On the minimaxness and coatomicness of local cohomology modules. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 1, pp. 177-190. doi: 10.21136/CMJ.2021.0383-20

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