Burch index, summands of syzygies and linearity in resolutions

TL;DR

This paper introduces the Burch index, an invariant of local rings, showing that its positivity indicates linearity in module resolutions and analyzing its values in specific cases.

Contribution

It defines the Burch index and demonstrates its implications for the structure of syzygies and resolutions in local rings, providing computational insights.

Findings

Burch index positivity implies linearity in resolutions.

For depth zero rings with Burch index ≥ 2, certain syzygies contain the residue field.

Computed Burch index in various classes of local rings.

Abstract

The Burch index is a new invariant of a local ring $R$ whose positivity implies a kind of linearity in resolutions of $R$-modules. For example, if $R$ has depth zero and Burch index at least $2$, then any non-free 7th $R$-syzygy contains the residue field as a direct summand. We also compute the Burch index in various cases of interest.