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 ${\mathbb{P}}^{n-1}$ and a proof of the conjecture in the case where $S=k[t]$.