  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.
