Boij-Söderberg conjectures for differential modules

Boij-Söderberg theory gives a combinatorial description of the set of Betti tables belonging to finite length modules over the polynomial ring $S=k[x_1,\dots ,x_n]$. We posit that a similar combinatorial description can be given for analogous numerical invariants of graded differential S-modules, which are natural generalizations of chain complexes. We prove several results that lend evidence in support of this conjecture, including a categorical pairing between the derived categories of graded differential S-modules and coherent sheaves on $P^{n-1}$ and a proof of the conjecture in the case where $S=k\left[t\right]$.