Finiteness Properties of Formal Local Cohomology Modules and Cohen–Macaulayness

Abstract

Let 𝔞 be an ideal of a local ring (R, 𝔪) and M a finitely generated R-module. We investigate the structure of the formal local cohomology modules , i ≥ 0. We prove several results concerning finiteness properties of formal local cohomology modules which indicate that these modules behave very similar to local cohomology modules. Among other things, we prove that if dim R ≤ 2, or either \bm𝔞 is principal or dim R/\bm𝔞 ≤1, then is Artinian for all i and j. Also, we examine the notion fgrade(\bm𝔞, M), the formal grade of M with respect to \bm𝔞 (i.e., the least integer i such that ). As applications, we establish a criterion for Cohen–Macaulayness of M, and also we provide an upper bound for cohomological dimension of M with respect to \bm𝔞.

Key Words:

• Coassociated prime ideals
• Cofinite modules
• Cohen–Macaulay modules
• Cohomological dimension
• Formal grade
• Formal local cohomology
• Generalized local cohomology
• Ring of invariants

2000 Mathematics Subject Classification:

• 13D45
• 13C14
• 13Exx

ACKNOWLEDGMENTS

In an earlier version, we have used a spectral sequence argument for the proof of Theorem 4.1. Professor Peter Schenzel has pointed out to us that they can also be deduced from [26, Theorem 3.5]. Here, we exposed his proposed argument in place of our original one. The authors would like to express their thanks to him for his valuable comments.

The second author was supported by a grant from IPM (No. 87130114).

Notes

Communicated by A. Singh.