Cominimaxness of local cohomology modules

Moharram Aghapournahr

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 1, page 75-86
  • ISSN: 0011-4642

Abstract

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Let R be a commutative Noetherian ring, I an ideal of R . Let t ∈ ℕ 0 be an integer and M an R -module such that Ext R i ( R / I , M ) is minimax for all i ≤ t + 1 . We prove that if H I i ( M ) is FD ≤ 1 (or weakly Laskerian) for all i < t , then the R -modules H I i ( M ) are I -cominimax for all i < t and Ext R i ( R / I , H I t ( M ) ) is minimax for i = 0 , 1 . Let N be a finitely generated R -module. We prove that Ext R j ( N , H I i ( M ) ) and Tor j R ( N , H I i ( M ) ) are I -cominimax for all i and j whenever M is minimax and H I i ( M ) is FD ≤ 1 (or weakly Laskerian) for all i .

How to cite

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Aghapournahr, Moharram. "Cominimaxness of local cohomology modules." Czechoslovak Mathematical Journal 69.1 (2019): 75-86. <http://eudml.org/doc/294842>.

@article{Aghapournahr2019,
abstract = {Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$. Let $t\in \mathbb \{N\}_0$ be an integer and $M$ an $R$-module such that $\{\rm Ext\}^i_R(R/I,M)$ is minimax for all $i\le t+1$. We prove that if $H^\{i\}_\{I\}(M)$ is $\{\rm FD\}_\{\le 1\}$ (or weakly Laskerian) for all $i<t$, then the $R$-modules $H^\{i\}_\{I\}(M)$ are $I$-cominimax for all $i<t$ and $\{\rm Ext\}^i_R(R/I,H^\{t\}_\{I\}(M))$ is minimax for $i=0,1$. Let $N$ be a finitely generated $R$-module. We prove that $\{\rm Ext\}^j_R(N,H^\{i\}_\{I\}(M))$ and $\{\rm Tor\}^R_\{j\}(N,H^\{i\}_I(M))$ are $I$-cominimax for all $i$ and $j$ whenever $M$ is minimax and $H^\{i\}_\{I\}(M)$ is $\{\rm FD\}_\{\le 1\}$ (or weakly Laskerian) for all $i$.},
author = {Aghapournahr, Moharram},
journal = {Czechoslovak Mathematical Journal},
keywords = {local cohomology; $\{\rm FD\}_\{\le n\}$ modules; cofinite modules; cominimax modules},
language = {eng},
number = {1},
pages = {75-86},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Cominimaxness of local cohomology modules},
url = {http://eudml.org/doc/294842},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Aghapournahr, Moharram
TI - Cominimaxness of local cohomology modules
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 1
SP - 75
EP - 86
AB - Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$. Let $t\in \mathbb {N}_0$ be an integer and $M$ an $R$-module such that ${\rm Ext}^i_R(R/I,M)$ is minimax for all $i\le t+1$. We prove that if $H^{i}_{I}(M)$ is ${\rm FD}_{\le 1}$ (or weakly Laskerian) for all $i<t$, then the $R$-modules $H^{i}_{I}(M)$ are $I$-cominimax for all $i<t$ and ${\rm Ext}^i_R(R/I,H^{t}_{I}(M))$ is minimax for $i=0,1$. Let $N$ be a finitely generated $R$-module. We prove that ${\rm Ext}^j_R(N,H^{i}_{I}(M))$ and ${\rm Tor}^R_{j}(N,H^{i}_I(M))$ are $I$-cominimax for all $i$ and $j$ whenever $M$ is minimax and $H^{i}_{I}(M)$ is ${\rm FD}_{\le 1}$ (or weakly Laskerian) for all $i$.
LA - eng
KW - local cohomology; ${\rm FD}_{\le n}$ modules; cofinite modules; cominimax modules
UR - http://eudml.org/doc/294842
ER -

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