Some results on top local cohomology and top formal local cohomology modules

ABSTRACT

Let (R,𝔪) be a complete noetherian local ring and M a finitely generated R-module of dimension n and 𝔞 an ideal of R.

At first, we find a relation between top local cohomology and top formal local cohomology modules. In fact, we show that there exists an ideal 𝔟 such that $H_{𝔞}^n\left(M\right)\simeq {𝔉}_{𝔟}^n\left(M\right)$ and there exists an ideal 𝔠 such that ${𝔉}_{𝔞}^n\left(M\right)\simeq H_{𝔠}^n\left(M\right)$. Several applications of this result are given. Among other things, it is shown that ${Att}_R\left(H_{𝔞}^n\right(M\left)\right)=MinV\left({Ann}_R{H}_{𝔞}^n\right(M\left)\right)={Ass}_R(R∕{Ann}_R{H}_{𝔞}^n(M\left)\right)$ and ${Att}_R\left({𝔉}_{𝔞}^n\right(M\left)\right)=MinV\left({Ann}_R{𝔉}_{𝔞}^n\right(M\left)\right)={Ass}_R(R∕{Ann}_R{𝔉}_{𝔞}^n(M\left)\right)$.

KEYWORDS:

• Attached primes
• formal local cohomology
• local cohomology

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D45
• 13E15

Acknowledgment

The author would like to thank the referee for his/her useful suggestions.