| eBook ISBN: | 978-1-4704-0476-5 |
| Product Code: | MEMO/186/872.E |
| List Price: | $70.00 |
| MAA Member Price: | $63.00 |
| AMS Member Price: | $42.00 |
| eBook ISBN: | 978-1-4704-0476-5 |
| Product Code: | MEMO/186/872.E |
| List Price: | $70.00 |
| MAA Member Price: | $63.00 |
| AMS Member Price: | $42.00 |
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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 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.
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Table of Contents
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Chapters
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Part 1. Nonexistence and existence
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1. Introduction
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2. Numerical conditions
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3. Homological methods
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4. Some refinements
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5. Constructing Artinian level algebras
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6. Constructing level sets of points
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7. Expected behavior
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Part 2. Appendix: A classification of codimension three level algebras of low socle degree
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Appendix A. Introduction and notation
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Appendix B. Socle degree 6 and type 2
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Appendix C. Socle degree 5
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Appendix D. Socle degree 4
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Appendix E. Socle degree 3
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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 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.
-
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
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Appendix E. Socle degree 3
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Appendix F. Summary
