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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, Notre Dame, IN
Yong Su Shin Sungshin Women’s University, Seoul, Korea
Front Cover for The Hilbert Function of a Level Algebra
Available Formats:
Electronic 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
Front Cover for The Hilbert Function of a Level Algebra
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  • Front Cover for The Hilbert Function of a Level Algebra
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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, Notre Dame, IN
Yong Su Shin Sungshin Women’s University, Seoul, Korea
Available Formats:
Electronic 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
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1862007; 139 pp
    MSC: 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.

  • 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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Volume: 1862007; 139 pp
MSC: 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.

  • 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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