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
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
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,
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)
** This work was supported by two grants from Shikoku University and Hokkaido University
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.