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

TL;DR

This paper proves a special case of the Buchsbaum-Eisenbud-Horrocks rank conjecture for certain graded modules using Boij-Soederberg theory, and provides asymptotic bounds for Betti numbers of ideal powers.

Contribution

It establishes the conjecture for modules with small regularity relative to their first syzygy degree using Boij-Soederberg theory.

Findings

The conjecture holds for modules with small regularity.

Provides asymptotic lower bounds for Betti numbers of ideal powers.

Demonstrates the application of Boij-Soederberg theory to rank conjectures.

Abstract

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.