Divisors on graphs, binomial and monomial ideals, and cellular resolutions

TL;DR

This paper explores the algebraic and combinatorial structures of ideals associated with graphs, revealing their resolutions and Betti tables, and providing proofs for several conjectures in the field.

Contribution

It introduces a unified approach to describe minimal cellular resolutions of binomial and monomial ideals related to graphs, connecting them through potential theory and lattice decompositions.

Findings

Resolutions of these ideals are closely related and have identical Betti tables.

Provides conceptual proofs for conjectures by Postnikov-Shapiro, Manjunath-Sturmfels, and Perkinson-Perlman-Wilmes.

Extends techniques to chip-firing games and related graph theories.

Abstract

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their $\mathbb{Z}$-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.