On Golod Subdeterminantal Ideals

TL;DR

This work characterizes when quotients by $2$-subdeterminantal ideals $I$ of a matrix $X$ yield Golod rings $S/I$. It establishes equivalences among Golodness, the existence of a linear resolution for $I$, and $I$ being generated by the $2\times 2$ minors of a single $2\times \ell$ (or $\ell\times 2$) submatrix $Y$ of $X$, equivalently the triviality of the product on the Koszul homology of $S/I$. The authors develop a tensor-product framework and leverage the Künneth theorem, Poincaré algebra properties, and base-case analyses ($2\times n$, $3\times 3$) to prove the main theorem and derive non-Golodness results for certain disjoint sums of $2$-subdeterminantal ideals. They apply the results to binomial edge ideals, yielding a complete criterion: a binomial edge ideal is Golod exactly when both graphs are complete and one is $K_2$, while highlighting that for $t\ge 3$ the Golod property does not force a linear resolution or a submatrix-determined form. The work blends homological algebra with determinantal combinatorics and provides computational evidence for higher-minor cases.

Abstract

Let $X=(x_{ij})_{m\times n}$ be a matrix of indeterminates and let $S=\mathbb{k}[x_{ij} \mid 1\leq i\leq m,\ 1\leq j\leq n]$ be a polynomial ring over an infinite field $\mathbb{k}$. Let $I$ be an ideal generated by a subset of the set of all $2\times2$ minors of $X$. We show that the quotient ring $S/I$ is Golod if and only if $I=I_2(Y)$ for some $2\times \ell$ or $\ell\times2$ submatrix $Y$ of $X$. In fact, we prove that Golodness of $S/I$ is equivalent to the triviality of the product on the Koszul homology of $S/I$ and to $I$ having a linear resolution. Along the way, we also prove a result on the non-Golodness of tensor products of rings under certain conditions.

Theorems & Definitions (20)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Proposition 3.1
• proof
• Proposition 3.2