Let $R$ be a polynomial ring over an algebraically closed field and let $A$ be a standard graded Cohen-Macaulay 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.
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.