Browsing by Author "Mgani, Damas Karmel"

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    On Hilbert functions and H-vectors of graded modules for finite sets of points in projective space
    (University of Dar es Salaam, 2018) Mgani, Damas Karmel
    In this research, we study the Hilbert functions and h-vectors of graded modules with support on finite sets of points in projective space, P_k^n To attain this, we construct the graded modules from the sets of points in projective space. For example, taking X as the set of points, we define an ideal Ix to be the homogeneous ideal in R generated by all forms vanishing at all points of X, and RX: = R/Ix the homogeneous coordinate ring of X. We use a computer software package for algebraic computations Macaulay 2 to study the Hilbert functions, h-vectors and the resulting Betti diagrams of the constructed graded modules. We then concentrate on proving the following three main results. First, we prove that the degree of a homogeneous ideal J for which RX/J is Artinian is the initial degree of the minimal generator(s) of J. This is done by studying the Hilbert function of a homogeneous coordinate ring RX/J. Apart from an ideal J we construct monomial ideals I, I* ⊆ RX then we investigate the structure of the resulting quotient rings. In addition, we prove that a submodule of torsion less module is torsion less. Second, we study the relationship between h-vectors of graded modules and structure of the associated Betti diagrams. Lastly, we present some characterizations of torsion less and reflexive modules over Noetherian rings and integral domains.