Betti numbers of graded modules with support on reduced sets of points in projective space

Let be a polynomial ring of variables over an algebraically closed field, and be a set of points in a projective space. Denote by the ideal of all functions vanishing on, and the homogeneous coordinate ring of the points. In this thesis we prove three main results. First, we prove that the -modules of homomorphism are isomorphic to some homogeneous ideal in. Second, we prove that the method of linkage gives the lowest initial degree of embedding of the canonical module as a canonical ideal for sets of points in. Lastly, we give a method of embedding the canonical module in least initial degree of the canonical ideal for general sets of points in