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The Hilbert Function of a Level Algebra
Anthony V. Geramita, Queen's University, Kingston, ON, Canada, Tadahito Harima, Hokkaido University of Education, Kushiro, Hokkaido, Japan, Juan C. Migliore, University of Notre Dame, IN, and Yong Su Shin, Sungshin Women's University, Seoul, Korea
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Memoirs of the American Mathematical Society
2007; 139 pp; softcover
Volume: 186
ISBN-10: 0-8218-3940-3
ISBN-13: 978-0-8218-3940-9
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 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