Beyond Numerics: the Existence of Pure Filtrations

A recent result of Boij-Söderberg and Eisenbud-Schreyer proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. We consider the follow-up question of whether this numerical decomposition ever corresponds to an actual filtration of the minimal free resolution itself. Our main result is an affirmative answer to this question in many surprising cases. As applications of our technique, we also obtain new results about the semigroup of Betti diagrams and about very singular spaces of matrices.