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|Paper IPM / M / 11464||
Let \fa be an ideal of a local ring (R,\fm) and M a finitely generated R-module. We investigate the structure of the formal local cohomology modules \vplnHi\fm(M/\fan M), 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 dimR ≤ 2 or either \fa is principal or dimR/\fa ≤ 1, then \TorjR(R/\fa,\vplnHi\fm(M/\fan M)) is Artinian for all i and j. Also, we examine the notion \fgrade(\fa,M), the formal grade of M with respect to \fa (i.e. the least integer i such that \vplnHi\fm(M/\fan M) ≠ 0). 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 \fa.
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