, Castelnuovo - Mumford Regularity , and Generic Initial Ideals

KOSZUL ALGEBRAS, CASTELNUOVO-MUMFORD REGULARITY, AND GENERIC INITIAL IDEALS Giulio Caviglia The University of Kansas Advisor: Craig Huneke August, 2004 The central topics of this dissertation are: Koszul Algebras, bounds for the Castelnuovo Mumford regularity, and methods involving the use of generic changes of coordinates and generic hyperplane restrictions. We give an introduction to Koszul algebras and prove some criteria to show that an algebra is Koszul. We use these methods to show that the Pinched Veronese, i.e. the toric ring defined as K[X3,X2Y,XY 2,Y 3,X2Z,Y 2Z,XZ2,Y Z2,Z3], is Koszul. The middle chapters are devoted to the Castelnuovo-Mumford regularity. We give a collection of techniques and formulas to compute the regularity by using hyperplane sections. For example we obtain some variations of a criterion due to Bayer and Stillman and a formula for the regularity that involves the postulation numbers. We study the combinatorial properties of a special kind of monomial ideal that we call weakly stable. We employ them to give a uniform bound, depending on the degree of the generators, for the regularity of all the homogeneous ideals in a polynomial ring. We also provide bounds for the regularity of the tensor product and Hom of two modules.