Finiteness properties of formal local cohomology modules and Cohen-Macaulayness

Let $\text{fa}$ be an ideal of a local ring $(R,\text{fm})$ and $M$ a finitely generated $R$-module. We investigate the structure of the formal local cohomology modules ${\text{vpl}}_n{H}_{\text{fm}}^i(M\mathrm{/}{\text{fa}}^{n}M)$, $i\ge 0$. 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 $\dim R\le 2$ or either $\text{fa}$ is principal or $\dim R/\fa\leq 1$,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 $\text{fgrade}(\text{fa},M)$, the formal grade of $M$ with respect to $\text{fa}$ (i.e. the least integer $i$ such that ${\text{vpl}}_n{H}_{\text{fm}}^i(M\mathrm{/}{\text{fa}}^{n}M)\mathrm{\ne }0$). 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 $\text{fa}$.