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.Comment: 18 pages, 2 figures, Minor revision