Linear strands of edge ideals of multipartite uniform clutters

TL;DR

This paper constructs the first linear strand of minimal free resolutions for edge ideals of d-partite d-uniform clutters, providing explicit descriptions and linking algebraic invariants to simplicial complexes.

Contribution

It introduces the first linear strand construction for these edge ideals and characterizes when they have linear resolutions, generalizing known results.

Findings

The first linear strand is supported on a relative simplicial complex.

Explicit minimal free resolutions are provided for ideals with linear resolutions.

Lyubeznik numbers relate to Betti numbers of certain simplicial complexes.

Abstract

We construct the first linear strand of the minimal free resolutions of edge ideals of $d$-partite $d$-uniform clutters. We show that the first linear strand is supported on a relative simplicial complex. In the case that the edge ideals of such clutters have linear resolutions, we give an explicit and surprisingly simple description of their minimal free resolutions, generalizing the known resolutions for edge ideals of Ferrers graphs and hypergraphs and co-letterplace ideals. As an application, we show that the Lyubeznik numbers that appear on the last column of the Lyubeznik table of the cover ideal of such clutters are Betti numbers of certain simplicial complexes. Furthermore, we restate a characterization for edge ideals of $d$-partite $d$-uniform clutters which have linear resolutions based on the recent characterization of arithmetically Cohen-Macaulay sets of points in a multiprojective space.