A special case of the Buchsbaum-Eisenbud-Horrocks rank conjecture

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.

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Published 1 January 2010