The differential-module Boij–Söderberg decomposition conjecture

Let $S=mathbbm{k}[x_1,\ldots,x_n]$ be a polynomial ring over a field $mathbbm{k}$. A degree sequence is the sequence of generating degrees of a pure resolution, and folding such a resolution in degree $0$ gives a Betti vector of a degree $0$ differential module. Differential-module Boij–Söderberg conjecture. Every differential module over $S$ with finite-length homology has a Betti vector that can be expressed as a positive rational combination of Betti vectors of folds of pure resolutions with finite-length homology whose degree sequences form a chain. This is a proposed Boij–Söderberg-type description for differential modules; the preceding results establish lower bounds and the one-variable case, but the general decomposition remains conjectural.