Memoirs of the American Mathematical Society 2007; 139 pp; softcover Volume: 186 ISBN10: 0821839403 ISBN13: 9780821839409 List Price: US$70 Individual Members: US$42 Institutional Members: US$56 Order Code: MEMO/186/872
 Let \(R\) be a polynomial ring over an algebraically closed field and let \(A\) be a standard graded CohenMacaulay quotient of \(R\). The authors state 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? The authors 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. The authors give new methods for showing that certain functions are NOT possible as the Hilbert function of a level algebra and also give new methods to construct level algebras. In a (rather long) appendix, the authors apply their 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. Table of Contents Part 1. Nonexistence and Existence  Introduction
 Numerical conditions
 Homological methods
 Some refinements
 Constructing Artinian level algebras
 Constructing level sets of points
 Expected behavior
Part 2. Appendix: A Classification of Codimension Three Level Algebras of Low Socle Degree  Appendix A. Introduction and notation
 Appendix B. Socle degree \(6\) and Type \(2\)
 Appendix C. Socle degree \(5\)
 Appendix D. Socle degree \(4\)
 Appendix E. Socle degree \(3\)
 Appendix F. Summary
 Appendix. Bibliography
