A criterion for isomorphism of Artinian Gorenstein algebras

Abstract

Let $A$ be an Artinian Gorenstein algebra over an infinite field~$k$ of characteristic either 0 or greater than 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 following surprising criterion has been obtained: two Artinian Gorenstein algebras $A$, $\widetilde {A}$ are isomorphic if and only if any two hypersurfaces $S_{\pi }$ and $S_{\tilde {\pi }}$ arising from $A$ and $\widetilde {A}$, 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.