A study of cofiniteness through minimal associated primes

Abstract

In this paper we shall investigate the concepts of cofiniteness of local cohomology modules and Abelian categories of cofinite modules over arbitrary Noetherian rings. Then we shall improve some of the results given in [3,5,9,10,11,13,17,19,20,21,22,26,28].

A Note on Abelian Categories of Cofinite Modules

“…This completes the inductive step in the construction. Now, we are ready to state and prove the main result of this article, which answers affirmatively[1, Question D].…”

“…The main purpose of this section is to prove Theorem 2.9, which presents an affirmative answer to a question raised by the present author in [1]. The following auxiliary lemmas are quite useful in the proof of Theorem 3.9.…”

“…We recall that the present author in [1], for any ideal I of R and any finitely generated R-module M, defined: Whether C (R, I) cof is Abelian?…”

Cofiniteness over Noetherian complete local rings

“…Concerning the Question 1, there are several papers in the literature containing some sufficient conditions for the ideals of R being in I (R), (see [2,3,4,6,7,8,17,20,23,26]). In section 3 of this paper, in order to finding a necessary and sufficient condition for the ideals of any Noetherian complete local ring (R, m) being in I (R), we will focus on the results of the more recently published article [2].…”

“…We recall that the present author in Ref. [2], for any ideal I of R and any finitely generated R-module M, defined: He proved that Question A has an affirmative answer in general if and only if Question B has so, (see [2,Proposition 4.7]).…”