Combinatorial Interpretations of some Boij-S\"oderberg Decompositions

TL;DR

This paper provides combinatorial interpretations of Boij-S"oderberg decompositions for specific classes of rings and modules, linking algebraic invariants to Ferrers hypergraphs and simplicial polytopes.

Contribution

It introduces combinatorial models for Betti table decompositions in the context of ideals with linear resolutions, Gorenstein rings, and quasi-Gorenstein modules.

Findings

Interpretations of Boij-S"oderberg coefficients via Ferrers hypergraphs and simplicial polytopes

Structural results on decompositions for quasi-Gorenstein modules

Applications to ideals with d-linear resolutions and their quotient rings

Abstract

Boij-S\"oderberg theory shows that the Betti table of a graded module can be written as a liner combination of pure diagrams with integer coefficients. Using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules.