Throughout this paper, we assume that (R;m) is a commutative Noetherian local ring with non-zero identity and a an ideal of R. For an integer i and a nitely generated R-module M let Hi a (M) denote the local cohomology module of M with respect to a (see [3] for the basic de nitions). Huneke [11] asked the question: When the modules Hi a(M) are Artinian. In general, this question is not true see for example [14] and [10], also the question is still true in many situations (see [16], [6], [13] and [1]). Recently Schenzel [19] has examined the structure of the modules lim n Hi m(M=anM) extensively. For each i, he called Fi a(M) := lim n Hi m(M=anM) the ith formal local cohomology module of M with respect to a. Not so much is known about these modules. In the case of a regular local ring they have been studied by Peskine and Szpiro (cf. [18, Chapter III]) in relation to the vanishing of local cohomology modules. Another kind of investigations about formal cohomology has been done by Faltings (cf. [8]). For more details on the notion of formal cohomology, we refer the reader to [12], [2] and [7]. Now it is natural to ask the following question for the formal cohomology: When are the formal local cohomology modules Fi a(M) Artinian?
