Boij-S\"oderberg Conjectures for Differential Modules

TL;DR

This paper explores a conjecture extending Boij-S"oderberg theory to graded differential modules, providing evidence through categorical pairings and proving the conjecture for polynomial rings in one variable.

Contribution

It introduces a conjecture for a combinatorial description of invariants of differential modules and proves it in the case of a single-variable polynomial ring.

Findings

Evidence supporting the conjecture through categorical pairings.

Proof of the conjecture for $S = k[t]$, a polynomial ring in one variable.

Establishment of a framework linking differential modules and coherent sheaves.

Abstract

Boij-S\"oderberg theory gives a combinatorial description of the set of Betti tables belonging to finite length modules over the polynomial ring $S = k[x_1, \ldots, x_n]$. 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 $S = k[t]$.