A constructive approach to one-dimensional Gorenstein k-algebras

Let R be the power series ring or the polynomial ring over a field k and let I be an ideal of R. Macaulay proved that the Artinian Gorenstein kalgebras R/I are in one-to-one correspondence with the cyclic R-submodules of the divided power series ring Γ. The result is effective in the sense that any polynomial of degree s produces an Artinian Gorenstein k-algebra of socle degree s. In a recent paper, the authors extended Macaulay's correspondence characterizing the R-submodules of Γ in one-to-one correspondence with Gorenstein d-dimensional k-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein k-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the G-admissible submodules of Γ. Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed.