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.
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