Artinian Gorenstein algebras of embedding dimension four and socle degree three

TL;DR

This paper characterizes Artinian Gorenstein algebras of embedding dimension four and socle degree three, showing they can be constructed via doubling from certain grade three perfect ideals, with explicit resolutions.

Contribution

It provides a complete description of these Gorenstein ideals as doubles of grade three perfect ideals and details their minimal free resolutions.

Findings

All such Gorenstein ideals are obtained by doubling from a grade three perfect ideal.

The minimal free resolution of the algebra can be explicitly described.

The construction involves embedding a shifted canonical module into the ideal.

Abstract

We prove that in the polynomial ring $Q=\mathsf{k}[x,y,z,w]$, with $\mathsf{k}$ an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals $I$ such that $(x,y,z,w)^4\subseteq I \subseteq (x,y,z,w)^2$ can be obtained by \emph{doubling} from a grade three perfect ideal $J\subset I$ such that $Q/J$ is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the $Q$-module $Q/I$ can be completely described in terms of a graded minimal free resolution of the $Q$-module $Q/J$ and a homogeneous embedding of a shift of the canonical module $\omega_{Q/J}$ into $Q/J$.