Finiteness properties of formal local cohomology modules and Cohen-Macaulayness

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 ${vpl}_nH^i_{fm}(M/fa^n M)$, $igeq 0$. 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 $dim Rleq 2$ or either $fa$ is principal or $dim R/faleq 1$, then $Tor_j^R(R/fa,{vpl}_nH^i_{fm}(M/fa^n M))$ 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 ${vpl}_nH^i_{fm}(M/fa^n M) eq 0$). 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 $fa$.