The classical Laplacian matrix of a graph G describes the combinatorial dynamics of the Abelian Sandpile Model and the more general Riemann-Roch theory of G.  The lattice ideal associated to the Laplacian provides a more recently considered algebraic perspective on this (re)emerging field.  The Laplacian ideal has a distinguished monomial initial ideal that has also been studied in connection with G-parking functions.  We study homological properties and show that a minimal free resolution of the initial ideal is supported on the bounded subcomplex of a section of the graphical arrangement of G.  As a corollary we obtain a combinatorial characterization of the Betti numbers in terms of acyclic orientations.  This generalizes constructions from Postnikov and Shaprio (for the case of the complete graph) and connects to work of Manjunath and Sturmfels, and Perkinson on the commutative algebra of Sandpiles.  This is joint work with Raman Sanyal.