Filtering free resolutions

TL;DR

This paper investigates when the Betti diagram decomposition of graded modules corresponds to actual free resolution filtrations, providing conditions and applications to resolve questions about free resolutions and quiver representations.

Contribution

It offers a sufficient condition linking Betti diagram decompositions to free resolution filtrations and applies this to resolve existence questions and study quiver representation semigroups.

Findings

Identifies when Betti diagram decompositions correspond to actual filtrations.

Shows non-existence of certain free resolutions with plausible Betti diagrams.

Analyzes the semigroup of quiver representations for the simplest wild quiver.

Abstract

A recent result of Eisenbud-Schreyer and Boij-S\"oderberg proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. When does this numerical decomposition correspond to an actual filtration of the minimal free resolution? Our main result gives a sufficient condition for this to happen. We apply it to show the non-existence of free resolutions with some plausible-looking Betti diagrams and to study the semigroup of quiver representations of the simplest "wild" quiver.