  The Buchsbaum-Eisenbud-Horrocks rank conjecture proposes lower bounds for the
Betti numbers of a graded module M based on the codimension of M. We prove a
special case of this conjecture via Boij-Soederberg theory. More specifically,
we show that the conjecture holds for graded modules where the regularity of M
is small relative to the minimal degree of a first syzygy of M. Our approach
also yields an asymptotic lower bound for the Betti numbers of powers of an
ideal generated in a single degree.
