Cohomological splitting, realization, and finiteness

We search for some splitting (resp. finiteness) criteria of a given module $M$ over a local ring $(R,\fm,k)$ in terms of the splitting (resp. finiteness) property of certain cohomological functors evaluated at $M$. In particular, we deal with the cohomological splitting question posted by Vasconcelos. We present a connection from our approach to the realization problem of Nunke. This is equipped with several applications. For instance, we recover some results of Jensen (and others) by applying simple methods. Additional applications, including a computation of the projective dimension of some injective modules, are given. This enables us to extend some results of Matlis (resp. Osofsky) on the projective dimension of $E_R(k)$ (resp. $\mathcal{Q}$) from Cohen-Macaulay rings (resp. regular rings) to non-Cohen-Macaulay (resp. Cohen-Macaulay) rings.