On a new criterion for isomorphism of Artinian Gorenstein algebras

TL;DR

This paper presents an algebraic proof of a criterion for isomorphism of Artinian Gorenstein algebras based on the affine equivalence of associated hypersurfaces, and relates these hypersurfaces to Macaulay's inverse systems.

Contribution

It provides a short algebraic proof of the hypersurface criterion for Gorenstein algebra isomorphism and connects the associated polynomials to inverse systems.

Findings

Two Gorenstein algebras are isomorphic iff their hypersurfaces are affinely equivalent.

The polynomial maps $P_{\pi}$ relate to Macaulay's inverse systems.

Restrictions of $P_{\pi}$ serve as inverse systems for the algebra.

Abstract

To every Gorenstein algebra $A$ of finite vector space dimension greater than 1 over a field $\FF$ of characteristic zero, and a linear projection $\pi$ on its maximal ideal ${\mathfrak m}$ with range equal to the annihilator $\Ann({\mathfrak m})$ of ${\mathfrak m}$, one can associate a certain algebraic hypersurface $S_{\pi}\subset{\mathfrak m}$, which is the graph of a polynomial map $P_{\pi}:\ker\pi\ra\Ann({\mathfrak m})\simeq\FF$. Recently, in {\rm\cite{FIKK}}, {\rm\cite{FK}} the following surprising criterion was obtained: two 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 CR-geometric argument. In the present paper we give a short algebraic proof of this statement. We also compare the polynomials $P_{\pi}$ with Macaulay's inverse systems. Namely, we show that the restrictions of $P_{\pi}$ to certain subspaces of $\ker\pi$ are inverse systems for $A$.