  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.
