A short proof of Bresinski's Theorem on Gorenstein semigroup rings generated by 4 elements

TL;DR

This paper provides a concise proof of Bresinski's Theorem, establishing that the defining ideal of certain Gorenstein semigroup rings generated by four elements is minimally generated by 3 or 5 elements, using Buchsbaum and Eisenbud's resolution theory.

Contribution

The paper offers a new, shorter proof of Bresinski's Theorem leveraging the structure theorem for Gorenstein rings of codimension 3.

Findings

The defining ideal of the semigroup ring is generated by 3 or 5 elements.

The proof simplifies previous arguments using minimal free resolutions.

The approach applies to symmetric semigroups generated by four elements.

Abstract

Let $H=\langle n_1, \ldots ,n_4\rangle$ be a numerical semigroup generated by $4$ elements, which is symmetric and let $k[H]$ be the semigroup ring of $H$ over a field $k$. H. Bresinski proved that the defining ideal of $k[H]$ is minimally generated by $3$ or $5$ elements. We give a new short proof of Bresinski's Theorem using the structure theorem of Buchsbaum and Eisenbud on the minimal free resolution of Gorenstein rings of embedding codimension $3$.