Mathematical Research Letters
Volume 17 (2010)
A special case of the Buchsbaum-Eisenbud-Horrocks rank conjecture
Pages: 1079 – 1089
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-Söderberg 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.