A Criterion for Isomorphism of Artinian Gorenstein Algebras

Abstract

Let $A$ be an Artinian Gorenstein algebra over an infinite field $k$ with either $char(k)=0$ or $char(k)>\nu$, where $\nu$ is the socle degree of $A$. To every such algebra and a linear projection $\pi$ on its maximal ideal $\mathfrak{m}$ with range equal to the socle $Soc(A)$ of $A$, one can associate a certain algebraic hypersurface $S_{\pi }\subset \mathfrak{m}$, which is the graph of a polynomial map $P_{\pi }:ker\pi \to Soc(A)\simeq k$. Recently, the author and his collaborators have obtained the following surprising criterion: two Artinian Gorenstein algebras $A$, $\tilde{A}$ are isomorphic if and only if any two hypersurfaces $S_{\pi }$ and $S_{\tilde{\pi }}$ arising from $A$ and $\tilde{A}$, respectively, are affinely equivalent. The proof is indirect and relies on a geometric argument. In the present paper we give a short algebraic proof of this statement. We also discuss a connection, established elsewhere, between the polynomials $P_{\pi }$ and Macaulay inverse systems.

To appear in the Journal of Commutative Algebra. arXiv admin note: substantial text overlap with arXiv:1201.6100