Filtering free resolutions

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.Comment: We correct a mistake in the proof of Corollary 4.2 in the published version of this paper. The mistake involves an incorrect definition for when two degree sequences are "sufficiently separated". The new definition weakens Theorem 1.3 somewhat, but the examples survive. We thank Amin Nematbakhsh and to Gunnar Floystad for bringing this mistake to our attention. We also correct some minor typo