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arXiv · 2212.03794

Boij-S\"oderberg Conjectures for Differential Modules

Maya Banks
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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 $S = k[t]$.

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Maya Banks. 2022-12-07. Boij-S\"oderberg Conjectures for Differential Modules. https://arxiv.org/abs/2212.03794

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