On the minimaxness and coatomicness of local cohomology modules

Hatamkhani, Marzieh, and Roshan-Shekalgourabi, Hajar. "On the minimaxness and coatomicness of local cohomology modules." Czechoslovak Mathematical Journal 72.1 (2022): 177-190. <http://eudml.org/doc/297955>.

@article{Hatamkhani2022,
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 $(H^i_I(M))_\{\mathfrak \{m\}\}$ is a minimax $R_\{\mathfrak \{m\}\}$-module for all $\mathfrak \{m\} \in \{\rm Max\} (R)$ 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 $i \ge n \ge 1$, then $H^i_I(M)$ is Artinian for $i \ge n$. 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 $i\ge n$), where $n\ge 1$, then they must be minimax. Consequently, a vanishing theorem is proved for local cohomology modules.},
author = {Hatamkhani, Marzieh, Roshan-Shekalgourabi, Hajar},
journal = {Czechoslovak Mathematical Journal},
keywords = {local cohomology module; minimax module; coatomic module; Artinian module; local-global principle},
language = {eng},
number = {1},
pages = {177-190},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the minimaxness and coatomicness of local cohomology modules},
url = {http://eudml.org/doc/297955},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Hatamkhani, Marzieh
AU - Roshan-Shekalgourabi, Hajar
TI - On the minimaxness and coatomicness of local cohomology modules
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 1
SP - 177
EP - 190
AB - 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 $(H^i_I(M))_{\mathfrak {m}}$ is a minimax $R_{\mathfrak {m}}$-module for all $\mathfrak {m} \in {\rm Max} (R)$ 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 $i \ge n \ge 1$, then $H^i_I(M)$ is Artinian for $i \ge n$. 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 $i\ge n$), where $n\ge 1$, then they must be minimax. Consequently, a vanishing theorem is proved for local cohomology modules.
LA - eng
KW - local cohomology module; minimax module; coatomic module; Artinian module; local-global principle
UR - http://eudml.org/doc/297955
ER -