On the structure of finitely generated modules and the unmixed degrees

Let $(R,m)$ be the homomorphic image of a Cohen-Macaulay local ring and M a finitely generated R-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every finitely generated R-module M is associated to a sequence of invariant modules. This module sequence expresses the deviation of M with the Cohen-Macaulay property. Our result generalizes the unmixed theorem of Cohen-Macaulayness for any finitely generated R-module. As an application we construct a new extended degree in the sense of Vasconcelos.