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Memoirs of the American Mathematical Society
2007; 139 pp; softcover
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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.
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