The linear strand of determinantal facet ideals

TL;DR

This paper constructs the linear strand of the resolution for determinantal facet ideals generated by maximal minors, linking it to clique complexes, and provides explicit formulas for Betti numbers and conditions for linear resolutions.

Contribution

It introduces a method to explicitly determine the linear strand of the resolution of determinantal facet ideals using clique complexes, and characterizes when these ideals have linear resolutions.

Findings

Explicit formulas for graded Betti numbers $eta_{i,i+m}(J)$.

Determined all sets $\\mathcal{S}$ for which $J$ has a linear resolution.

Connected the linear strand to the clique complex of the associated $m$-clutter.

Abstract

Let $X$ be an $(m\times n)$-matrix of indeterminates, and let $J$ be the ideal generated by a set $\mathcal{S}$ of maximal minors of $X$. We construct the linear strand of the resolution of $J$. This linear strand is determined by the clique complex of the $m$-clutter corresponding to the set $\mathcal{S}$. As a consequence one obtains explicit formulas for the graded Betti numbers $\beta_{i,i+m}(J)$ for all $i\geq 0$. We also determine all sets $\mathcal{S}$ for which $J$ has a linear resolution.