Vanishing of the Top Local Cohomology Modules over Noetherian Rings

Proceedings Mathematical Sciences

Kamran Divaani-Aazar¹ 

[1] Department of Mathematics, Az-Zahra University, Vanak, Post Code , Tehran, Iran$$

关键词: Artinian modules;    attached prime ideals;    cohomological dimension;    formally isolated;    local cohomology;    secondary representations.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let 𝑅 be a (not necessarily local) Noetherian ring and 𝑀 a finitely generated 𝑅-module of finite dimension 𝑑. Let $mathfrak{a}$ be an ideal of 𝑅 and $mathfrak{M}$ denote the intersection of all prime ideals $mathfrak{p}inmathrm{Supp}_R H^d_a(M)$. It is shown that$$H^d_a(M)simeq H^d_{mathfrak{M}}(M)/sumlimits_{ninmathbb{N}}langle mathfrak{M}angle(0:_{H^d_{mathfrak{M}}(M)}a^n),$$where for an Artinian 𝑅-module 𝐴 we put $langlemathfrak{M}angle A=cap_{ninmathbb{N}}mathfrak{M}^n A$. As a consequence, it is proved that for all ideals $mathfrak{a}$ of 𝑅, there are only finitely many non-isomorphic top local cohomology modules $H^d_a(M)$ having the same support. In addition, we establish an analogue of the Lichtenbaum–Hartshorne vanishing theorem over rings that need not be local.

【 授权许可】

Unknown