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.

Table of Contents

• Introduction
• Numerical conditions
• Homological methods
• Some refinements
• Constructing Artinian level algebras
• Constructing level sets of points
• Expected behavior

• 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