  Let \fa be an ideal of a local ring (R,\fm) and M a finitely generated
R-module. This paper concerns 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)\neq 0). We show that \fgrade(\fa,M)\geq \depth
M-\cd_{\fa}(M), and as a result, we establish a new characterization of
Cohen-Macaulay modules. As an application of this characterization, we show
that if M is Cohen-Macaulay and L a pure submodule of M with the same support
as M, then \fgrade(\fa,L)=\fgrade(\fa,M). Also, we give a generalization of the
Hochster-Eagon result on Cohen-Macaulayness of invariant rings.
