The Tor algebra of trimmings of Gorenstein ideals

TL;DR

This paper constructs explicit free resolutions for certain Gorenstein ideal trimings in a 3-dimensional regular local ring, analyzes their algebraic structure, and confirms conjectures about their Tor algebras.

Contribution

It provides explicit free resolutions and DG algebra structures for trimmed Gorenstein ideals, advancing understanding of their Tor algebra and confirming related conjectures.

Findings

Constructed explicit free resolutions for trimmed Gorenstein ideals.

Computed partial and full DG algebra structures on these resolutions.

Confirmed conjectures of Christensen, Veliche, and Weyman for ideals of class G.

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. Buchsbaum and Eisenbud proved 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 $I$. Building on a recent paper of Vandebogert, we construct an explicit free resolution of $R/J$ and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class $\mathbf{G}$ hold true in our context. Furthermore, we address the realizability question for ideals of class $\mathbf{G}$.