It is well known that there are non-linear Poisson equations in two- dimensions that admit infinitely many "higher" conservation laws. The characteristic cohomology theory developed by Bryant and Griffiths defines conservation laws of PDE's as universal cohomology classes on a target space, much the way that Chern classes of vector bundles can be defined as the pullbacks of cohomology classes on universal classifying spaces. Although the examples of the theory of characteristic cohomology have appeared in the literature, for the first time we completely determine the characteristic cohomlogy on the infinite prolongation in the case of the non-linear Poisson equation. As expected, we recover the well known infinite sequence of conservation laws, but now as cohomology classes on a target space. We rely on an interesting connection between formal Killing fields and conservation laws. We will conclude by discussing 1) a potential method for "linearization" of the nonlinear PDE that would parallel the theory of the spectral curve, 2) a conjecture about the rationality of the characteristic cohomology, and 3) the next steps in this line of investigation, namely, higher dimensional systems and a global, geometric approach to the conservation laws of the submanifold geometry governed by the PDE's in question.