On Golod Subdeterminantal Ideals

TL;DR

This paper characterizes when quotient rings by certain subdeterminantal ideals are Golod, linking Golodness to the ideal's structure, linear resolutions, and Koszul homology triviality.

Contribution

It provides a complete characterization of Golod subdeterminantal ideals generated by 2x2 minors, connecting algebraic properties to ideal structure.

Findings

Golodness of S/I occurs iff I is generated by a 2xℓ or ℓx2 submatrix minor

Golodness is equivalent to trivial product on Koszul homology

I has a linear resolution when S/I is Golod

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.