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.

Key Words:

• Cofinite modules
• FSF modules
• Minimal modules
• Weakly Laskerian modules
• nth finiteness dimensions

2010 Mathematics Subject Classification:

• 13D45
• 14B15
• 13E05

ACKNOWLEDGMENTS

The authors are deeply grateful to the referee for a very careful reading of the manuscript and many valuable suggestions and for drawing the authors’ attention to Examples 4.2, 4.3, and 4.4. Also, we would like to thank Professors Hossein Zakeri and Kamran Divaani-Aazar for their careful reading of the first draft and many helpful suggestions. Finally, the authors would like to thank the Institute for Research in Fundamental Sciences (IPM) for its financial support.

This research of the first and second authors was in part supported by a grant from IPM (No. 89130048, 89130053).

The third author has been supported by the Azarbaijan Shahid Madani University.

Notes

Communicated by S. Goto.

Dedicated to Professor Markus P. Brodmann