A constructive approach to one-dimensional Gorenstein $k$-algebras

TL;DR

This paper develops finite, constructive methods for building one-dimensional Gorenstein $k$-algebras, extending classical Macaulay correspondence and applying to linkage and semigroup rings.

Contribution

It introduces explicit procedures for constructing one-dimensional Gorenstein $k$-algebras, addressing the non-finite generation issue in higher dimensions.

Findings

Provides finite algorithms for Gorenstein algebra construction.

Extends Macaulay's correspondence to positive-dimensional cases.

Applies methods to Gorenstein linkage and affine semigroup rings.

Abstract

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 $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $\Gamma. $ 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 $\Gamma $ 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 $\Gamma. $ Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed.