Electronic ISBN:  9781470404765 
Product Code:  MEMO/186/872.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 186; 2007; 139 ppMSC: Primary 13; Secondary 14;
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

Chapters

Part 1. Nonexistence and existence

1. Introduction

2. Numerical conditions

3. Homological methods

4. Some refinements

5. Constructing Artinian level algebras

6. Constructing level sets of points

7. 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


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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.

Chapters

Part 1. Nonexistence and existence

1. Introduction

2. Numerical conditions

3. Homological methods

4. Some refinements

5. Constructing Artinian level algebras

6. Constructing level sets of points

7. 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