### Article

Kyungpook Mathematical Journal 2023; 63(1): 37-43

**Published online** March 31, 2023 https://doi.org/10.5666/KMJ.2022.63.1.37

Copyright © Kyungpook Mathematical Journal.

### On the Local Cohomology and Formal Local Cohomology Modules

Shahram Rezaei^{*} and Behruz Sadeghi

Department of Mathematics, Payame Noor University (PNU), Tehran, Iran

e-mail : sha.rezaei@gmail.com and b-sadeqi@student.pnu.ac.ir

**Received**: July 31, 2022; **Revised**: November 20, 2023; **Accepted**: January 17, 2023

Let 𝔞 and 𝔟 be ideals of a commutative Noetherian ring

**Keywords**: formal local cohomology, local cohomology

Throughout this paper,

The reader can refer to [3], for the basic properties of local cohomology.

Let

In [8], we studied local cohomology module

Here, we obtain some relations between

We determine the support of the top local cohomology module

A non-zero

Recall that for any

We need the following lemmas in the proof of main results.

**Lemma 2.1.** Let

**Lemma 2.2.** Let

The following theorem is a main result of [8] and has a key role in our proofs.

**Theorem 2.3.** Let

i)

ii)

ii) See [8, Theorem 2.15].

In the following, we obtain a generalization of [6, Theorem 2.6].

**Theorem 2.4.** Let

**Theorem 2.5.** Let

**Theorem 2.6.** Let

Similar to the above method, we can show that

and the proof is complete.

**Corollary 2.7.** Let

**Corollary 2.8.** Let

i)

ii)

Now, assume that

Now, the result (ii) follows immediately from [8, Corollary 2.16], as required.

In the above results, we saw that some basic properties of two

**Theorem 2.9.** Let

i)

ii)

iii)

If

Since

If

Now assume that

Since

In the remainder, we prove some results about

**Theorem 2.10.** Let

**Theorem 2.11.** Let

**Corollary 2.12.** Let

**Theorem 2.13.** Let

i)

ii)

ii) See [8, Theorem 3.13].

**Theorem 2.14.** Let

Now, assume that

**Theorem 2.15.** Let

**Theorem 2.16.** Let

Now we assume that

**Theorem 2.17.** Let

**Theorem 2.18 ** Let

### Acknowledgments.

The author would like to thank the referee for careful reading and many useful suggestions.

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