Czechoslovak Mathematical Journal, Vol. 72, No. 1, pp. 177-190, 2022
On the minimaxness and coatomicness of local cohomology modules
Marzieh Hatamkhani, Hajar Roshan-Shekalgourabi
Received September 7, 2020. Published online December 8, 2021.
Abstract: 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 $i \geq n \geq1$, 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 $n\geq1$, then they must be minimax. Consequently, a vanishing theorem is proved for local cohomology modules.
Affiliations: Marzieh Hatamkhani, Department of Mathematics, Faculty of Science, Arak University, Arak, 65183-5-5638, Iran, e-mail: m-hatamkhani@araku.ac.ir; Hajar Roshan-Shekalgourabi (corresponding author), Department of Basic Sciences, Arak University of Technology, P.O. Box 38135-1177, Arak, Iran, e-mail: hrsmath@gmail.com
