Simplicial complexes and Macaulay's inverse systems

Details Simplicial complexes and Macaulay's inverse systems Simplicial complexes and Macaulay's inverse systems

2007-12-11

arxiv.org/abs/0712.1804v2

Math. Z. 265 (2010), no. 1, 151-160

Mathematics

Commutative Algebra

Very minor changes. To appear in Math. Z

Scientific paper

Let $\Delta$ be a simplicial complex on $V = \{x_1,...,x_n\}$, with Stanley-Reisner ideal $I_{\Delta}\subseteq R = k[x_1,...,x_n]$. The goal of this paper is to investigate the class of artinian algebras $A=A(\Delta,a_1,...,a_n)= R/(I_{\Delta},x_1^{a_1},...,x_n^{a_n})$, 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 $\Delta$. 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 $A(\Delta,a_1,...,a_n)$ is a level algebra.

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