Burch Index, Summands of Syzygies and Linearity in Resolutions

Abstract

AbstractThe Burch index is a new invariant of a local ring R whose positivity implies a kind of linearity in resolutions of R-modules. We show that 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 compute the Burch index in various cases of interest.