A complex-analytic proof of a criterion for isomorphism of Artinian Gorenstein algebras

Abstract

Let A be an Artinian Gorenstein algebra over a field k of characteristic zero with dim k A > 1. To every such algebra and a linear projection π on its maximal ideal m with range equal to the socle Soc(A) of A, one can associate a certain algebraic hypersurface S π ⊂ m, which is the graph of a polynomial map P π : ker π → Soc(A) k. Recently, the following surprising criterion has been obtained: two Artinian Gorenstein algebras A and A are isomorphic if and only if any two hypersurfaces S π and S π arising from A and A, respectively, are affinely equivalent. In the present paper, we focus on the cases k = R and k = C and explain how in these situations the above criterion can be derived by complex-analytic methods. The complex-analytic proof for k = R has not previously appeared in print but is foundational for the general result. The purpose of our paper is to present this proof and compare it with that for k = C, thus highlighting a curious connection between complex analysis and commutative algebra.