Trimming Five Generated Gorenstein Ideals

Abstract

Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension 3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. It follows from a result of Buchsbaum and Eisenbud that there is a skew-symmetric matrix of odd size such that $I$ is generated by the sub-maximal pfaffians of this matrix. Let $J$ be the ideal obtained by multiplying some of the pfaffian generators of $I$ by $\mathfrak{m}$; we say that $J$ is a trimming of $I$. In a previous work, the first author and A. Hardesty constructed an explicit free resolution of $R/J$ and computed a DG algebra structure on this resolution. They utilized these products to analyze the Tor algebra of such trimmed ideals. Missing from their result was the case where $I$ is five generated. In this paper we address this case.