Free Resolutions and Sparse Determinantal Ideals

TL;DR

This paper extends the understanding of sparse determinantal ideals by computing their minimal free resolutions, showing they have linear resolutions and universal Gröbner bases, with projective dimension depending only on zero columns.

Contribution

Introduces a technique for pruning free resolutions to handle sparse matrices, and proves properties like linearity and universal Gröbner bases for these ideals.

Findings

Sparse determinantal ideals have linear resolutions over integers.

The projective dimension depends only on the number of zero columns.

Betti numbers are invariant under term order, and generators form a universal Gröbner basis.

Abstract

A sparse generic matrix is a matrix whose entries are distinct variables and zeros. Such matrices were studied by Giusti and Merle who computed some invariants of their ideals of maximal minors. In this paper we extend these results by computing a minimal free resolution for all such sparse determinantal ideals. We do so by introducing a technique for pruning minimal free resolutions when a subset of the variables is set to zero. Our technique correctly computes a minimal free resolution in two cases of interest: resolutions of monomial ideals, and ideals resolved by the Eagon-Northcott Complex. As a consequence we can show that sparse determinantal ideals have a linear resolution over the integers, and that the projective dimension depends only on the number of columns of the matrix which are identically zero. Finally, we show that all such ideals have the property that regardless of the term order chosen, the Betti numbers of the ideal and its initial ideal are the same. In particular the nonzero generators of these ideals form a universal Gr\"obner basis.