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 , … , 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 [ t ] .