Abstract Let R be a, polynomial ring over an algebraically closed field and let A be a standard graded Cohen-Macaulay quotient of R. We say that A is a level algebra if the last module in the minimal free resolution of A (as R-module) is of the form R(—s)a, where s and a are positive integers. When a = 1 these are also known as Gorenstein algebras. The basic question addressed in this paper is: What can be the Hilbert Function of a level algebra? Our approach is to consider the question in several particular cases. E.g. when A is an Artinian algebra, or when A is the homogeneous coordinate ring of a reduced set of points, or when A satisfies the Weak Lefschetz Property. We give new methods for showing that certain functions are NOT possible as the Hilbert function of a level algebra and we also give new methods to construct level algebras. In a (rather long) appendix, we apply our results to give complete lists of all possible Hilbert functions in the case that the codimension of A = 3, s is small and a takes on certain fixed values. Received by the editor January 26, 2005. 1991 Mathematics Subject Classification. Primary: 13D40, 13D02 Secondary: 13C13, 13C40, 14C20. Key words and phrases. Level algebras, Hilbert function, liaison, link-sum, Weak Lefschetz property, Betti diagram, minimal free resolution. * Supported in part by a grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada and by the Ministero dell'Istruzione, dell'Universita e della Ricerca (MURST) of Italy. ** This work was supported by two grants from Shikoku University and Hokkaido University of Education. t Part of this work was done while this author was sponsored by the National Security Agency under Grant Number MDA904-03-1-0071. •'•This work was supported by a grant from Sungshin Women's University in 2002.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2007 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.