On Hilbert functions and H-vectors of graded modules for finite sets of points in projective space
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Date
2018
Authors
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Publisher
University of Dar es Salaam
Abstract
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.
Description
Available in print form, Eat Africana Collection, Dr. Wilbert Chagula Library,(THS EAF QA150.T34M42)
Keywords
Algebra, graded modules, Finite group, Projective space, Tanzania
Citation
Mgani, D. K. (2018) On Hilbert functions and H-vectors of graded modules for finite sets of points in projective space. Masters dissertation, University of Dar es Salaam, Dar es Salaam.