Monomials, Binomials, and Riemann-Roch

TL;DR

This paper explores the algebraic and combinatorial structures underlying the Riemann-Roch theorem on graphs, focusing on lattice ideals, G-parking functions, and minimal free resolutions, with special attention to saturated graphs.

Contribution

It introduces a new algebraic framework connecting Riemann-Roch on graphs with monomial ideals and provides a detailed analysis of syzygies and resolutions for these ideals.

Findings

For saturated graphs, the ideals are generic and have a minimal free resolution given by the Scarf complex.

Syzygies can be obtained by degeneration in non-saturated cases.

Develops a Riemann-Roch theory for artinian monomial ideals.

Abstract

The Riemann-Roch theorem on a graph G is related to Alexander duality in combinatorial commutive algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When G is a saturated graph, these ideals are generic and the Scarf complex is a minimal free resolution. Otherwise, syzygies are obtained by degeneration. We also develop a self-contained Riemann-Roch theory for artinian monomial ideals.