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The Hilbert Function of a Level Algebra
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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 homogeneo...
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:
Numerical conditions
Homological methods
Some refinements
Constructing Artinian level algebras
Constructing level sets of points
Expected behavior
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
Table of Contents:
Numerical conditions
Homological methods
Some refinements
Constructing Artinian level algebras
Constructing level sets of points
Expected behavior
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