Research ArticleNo Access

Coartinianess of local homology modules for ideals of small dimension

Jingwen Shen

Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China

E-mail Address: shenjw0609@163.com

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Pinger Zhang

Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China

E-mail Address: 275661482@qq.com

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Xiaoyan Yang

Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China

E-mail Address: yangxy@nwnu.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0219498823502122Cited by:1 (Source: Crossref)

Abstract

Let $𝔞$ be an ideal of a commutative noetherian ring $R$ and $M$ an $R$ -module with Cosupport in $V (𝔞)$ . We show that $M$ is $𝔞$ -coartinian if and only if ${Ext}_{R}^{i} (R / 𝔞, M)$ is artinian for all $0 \leq i \leq cd (𝔞, M)$ , which provides finite steps to examine $𝔞$ -coartinianess. We also consider the duality of Hartshorne’s questions: for which rings $R$ and ideals $𝔞$ are the modules $H_{i}^{𝔞} (M)$ $𝔞$ -coartinian for every artinian $R$ -module $M$ and all $i \geq 0$ ; whether the category $𝒞 {(R, 𝔞)}_{coa}$ of $𝔞$ -coartinian modules is an abelian subcategory of the category of $R$ -modules, and establish affirmative answers to these questions in the case $cd (𝔞, R) \leq 1$ and $dim R / 𝔞 \leq 1$ .

Communicated by E. Jespers

Keywords:

AMSC: 13D45, 13E05

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