The main objects of study in this paper are local cohomology modules. We write Hi I(M) for the ith local cohomology of a module M with respect to some ideal I. In this thesis we will look only at the local cohomology of finitely generated modules unless otherwise stated. This cohomology theory gives invariants to help measure many important properties in commutative algebra as well as algebraic geometry. If we consider the spectrum of a ring as a scheme, or restrict attention to an affine subscheme of something larger, it is often easier to define sections or functions on an open subset then on the whole space. The local cohomology modules can be viewed as sheaf cohomology on the complement of the closed set cut out by the ideal involved. This means elements of the first local cohomology module represent obstructions to extending sections across the whole space. In particular, having the first local cohomology vanish means we can define sections on the open set away from the zero set of our ideal and they always extend to the whole scheme.
