| Let (R, P) be a commutative, Noetherian local ring with identity, let I, J be two ideals of R and m, n ∈ N . Also consider M, N two finite R-modules.;After a quick introduction to the subject and a brief review of the existing results on Hilbert functions associated to the Tor and Ext functors, we devote the first chapter to the gathering of the tools needed for the rest of this dissertation.;In the second chapter, we investigate the length functions associated to the modules Tori(M, N/ InN), Exti( M, N/InN) and Ext i(N/InN, M) in order to establish whether they are (eventually) polynomial or not and to give a degree estimate for that polynomial, whenever possible. The main result in this chapter is that, unlike the case of the first two modules mentioned above (which always have polynomial growth), in the case of Ext i(N/InN, M) polynomial growth occurs in the presence of a mild condition on supports, but not always. An example shedding some light on why polynomial growth may fail is included. In the case i = d and M = N = R , a Cohen-Macaulay local ring, the dominant term of this polynomial is identified.;In the third chapter, we turn our attention to the corresponding two-variable length functions associated to the modules Tori( N/JmN, M/InM ) and Exti(N/ JmN, M/InM). Under a simple support condition, we give necessary and sufficient conditions for the module Tori(N/J mN, M/InM) to have its length given by a polynomial, for m, n ≫ 0. In the classical case when M ⊗ N has finite length, we prove that polynomial growth always occurs and give a formula for the length of Tori( N/JmN, M/InM ). Some particular situations for Exti( N/JmN, M/InM ) are also investigated.;The last chapter presents a different approach to the question of vanishing of cohomology in injective complexes, compared to those usually taken when proving Grothendieck's classical Vanishing Theorem. This approach is based on filtering the given minimal injective resolution E of a module M by subcomplexes determined by the heights of the associated primes of the modules in E . Grothendieck's Vanishing Theorem is recovered as a particular case. |
