Hilbert -Samuel polynomials and building indecomposable modules

Let (R, [special characters omitted], k) be a Noetherian local ring and M and N be finitely generated. In this thesis, we give precise formulas for the generalized Hilbert-Samuel polynomials associated to the torsion and contravariant extension functors, that is, polynomials giving the lengths of the modules [special characters omitted] and [special characters omitted], respectively. One application of these results is that they can be used to give information about the dimensions of syzygies of finite length modules. We also show this if R is complete and has depth at least 2, then one can build indecomposable modules of arbitrarily prescribed constant rank. Moreover, if R is assumed to be Cohen-Macaulay, then these modules can be chosen to be maximal Cohen-Macaulay when localized on the punctured spectrum