Minimaxness and Cofiniteness Properties of Local Cohomology Modules

Abstract

Let R be a commutative Noetherian ring, I an ideal of R, and M a non-zero R-module. The purpose of this article is to introduce the notation of the nth finiteness dimension for all n ∈ ℕ0, and to prove the following results: (i) ; (ii) The R-modules are I-cofinite for all and for all minimax submodules N of , the R-modules are finitely generated, whenever is finite. This implies that if I has dimension one, then is I-cofinite for every i ≥ 0, which is a generalization of the main results of Delfino–Marley, Yoshida, and Bahmanpour–Naghipour. (iii) , whenever R is semilocal. (iv) The R-modules are weakly Laskerian for all j ≥ 0 and all , whenever (R, 𝔪) is a complete Noetherian local ring. Moreover, in this situation for all weakly Laskerian submodules N of , the R-modules are weakly Laskerian, whenever is finite. In addition, some examples about , for n = 0, 1, 2, 3, are included.