On the formal grade of finitely generated modules over local rings

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) eq 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.