# The Hilbert Function of a Level Algebra

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*Anthony V. Geramita; Tadahito Harima; Juan C. Migliore; Yong Su Shin*

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

# Table of Contents

## The Hilbert Function of a Level Algebra

- Contents v6 free
- Part 1. Nonexistence and Existence 18 free
- Part 2. Appendix: A Classification of Codimension Three Level Algebras of Low Socle Degree 7380