Results of Formal Local Cohomology Modules

Let (R, m) be a commutative Noetherian local ring, a an ideal of R, and M a finitely generated R-module. We show that for a non-negative integer t the following cases are equivalent: (a) The formal local cohomology modules lim(<- n) H-m(i)(M/a(n)M) are Artinian for all i < t; (b) a subset of Rad(Ann(lim(<- n)H(m)(i)(M/a(n)M))) for all i < t. If one of the above cases holds, then lim(<- n) H-m(i) (M/a(n)M)/alim(<- n)H(m)(i) (M/a(n)M) is Artinian. Also, there are some results concerning finiteness properties of formal local cohomology modules.