Betti numbers, Hilbert series and the strong Lefschetz property
Abstract
In this thesis we answer two questions relating to numerical invariants of rings and modules. In particular we present results on the Hilbert series and Betti numbers of finitely generated graded modules over a polynomial ring. In Chapter 2 we discuss a classical result of Sperner in extremal combinatorics on the maximal length of an antichain. We describe the relationship between Sperner's theorem and the commutative algebra questions considered in this thesis. In Chapter 3 we consider finite length graded modules over a polynomial ring that have the strong Lefschetz property. For such a module we determine necessary and sufficient conditions on its Hilbert series in order that tensor products with cyclic modules over a one dimensional polynomial ring also have the strong Lefschetz property. In Chapter 4 we utilize an algebraic version of Sperner's theorem to investigate the Betti numbers of homogeneous ideals in a polynomial ring that contain a monomial complete intersection. Given a fixed monomial complete intersection in a polynomial ring we determine the maximum first and last Betti numbers for homogeneous ideals that contain the complete intersection. In the case of a polynomial ring in three variable we also determine the maximum second Betti number.
Degree
Ph.D.
Advisors
Heinzer, Purdue University.
Subject Area
Applied Mathematics|Mathematics
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