On a new criterion for isomorphism of Artinian Gorenstein algebras

2012/01/30 by Isaev, A. V.
#13H10 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1201.6100

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 π 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_π⊂\mathfrak m, which is the graph of a polynomial map P_π:kerπ\ra\Ann(\mathfrak m)≃\FF. Recently, in \rm\citeFIKK, \rm\citeFK the following surprising criterion was obtained: two Gorenstein algebras A, A are isomorphic if and only if any two hypersurfaces S_π and S_π arising from A and 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_π with Macaulay's inverse systems. Namely, we show that the restrictions of P_π to certain subspaces of kerπ are inverse systems for A.

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