On Golod Subdeterminantal Ideals

Abstract

Let $X=(x_{ij}{)}_{m\times n}$ be a matrix of indeterminates and let $S=\mathbb{k}[x_{ij}\mid 1\le i\le m,1\le j\le n]$ be a polynomial ring over an infinite field $\mathbb{k}$. Let $I$ be an ideal generated by a subset of the set of all $2\times 2$ 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 $\ell \times 2$ 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.

11 pages. Comments welcome!