The structure of the inverse system of Gorenstein k-algebras

Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein $k$-algebras. To date a general structure for Gorenstein $k$-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given.