Boij-S\"oderberg Conjectures for Differential Modules
Maya Banks

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 . We posit that a similar combinatorial description can be given for analogous numerical invariants of graded differential -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 -modules and coherent sheaves on and a proof of the conjecture in the case where .
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Taxonomy
TopicsCommutative Algebra and Its Applications · Polynomial and algebraic computation · Algebraic structures and combinatorial models
