paper

Simplicial complexes and Macaulay's inverse systems

arXiv:0712.1804

Abstract

Let $I^$ be a simplicial complex on $V = {x_{1}, ..., x_{n}}$ , with Stanley-Reisner ideal $I_{I^} \subseteq R = k [x_{1}, ..., x_{n}]$ . The goal of this paper is to investigate the class of artinian algebras $A = A (I^, a_{1}, ..., a_{n}) = R / (I_{I^} , x_{1}^{a_{1}}, ..., x_{n}^{a_{n}})$ , where each $a_{i} \geq 2$ . By utilizing the technique of Macaulay's inverse systems, we can explicitly describe the socle of $A$ in terms of $I^$ . As a consequence, we determine the simplicial complexes, that we will call {\em levelable}, for which there exists a tuple $(a_{1}, ..., a_{n})$ such that $A (I^, a_{1}, ..., a_{n})$ is a level algebra.

Very minor changes. To appear in Math. Z